Ticket #9880: trac_9880-fix_doctests-sage_5_10_beta5.patch

File trac_9880-fix_doctests-sage_5_10_beta5.patch, 135.0 KB (added by jpflori, 8 years ago)

Updated patch to apply to 5.10.beta5 without fuzz

  • doc/de/tutorial/introduction.rst

    # HG changeset patch
    # User Burcin Erocal <burcin@erocal.org>
    # Date 1369307548 -7200
    # Node ID 43cc30a191f8e6b53562963439dbb7c255cef6ea
    # Parent  acf9da97197945d9f1e200b0c993423115b89d86
    Trac #9880: Fix doctests after order changes in Pynac.
    
    diff --git a/doc/de/tutorial/introduction.rst b/doc/de/tutorial/introduction.rst
    a b  
    4848    1
    4949
    5050    sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k
    51     1/(I*sqrt(3) + 5/9*sqrt(73) + 3/4)
     51    36/(20*sqrt(73) + 36*I*sqrt(3) + 27)
    5252    sage: N(k)
    5353    0.165495678130644 - 0.0521492082074256*I
    5454    sage: N(k,30)      # 30 "bits"
    5555    0.16549568 - 0.052149208*I
    5656    sage: latex(k)
    57     \frac{1}{i \, \sqrt{3} + \frac{5}{9} \, \sqrt{73} + \frac{3}{4}}
     57    \frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}
    5858
    5959.. _installation:
    6060
  • doc/de/tutorial/tour_algebra.rst

    diff --git a/doc/de/tutorial/tour_algebra.rst b/doc/de/tutorial/tour_algebra.rst
    a b  
    5454    sage: eq2 = q*y+p*x==-6
    5555    sage: eq3 = q*y^2+p*x^2==24
    5656    sage: solve([eq1,eq2,eq3,p==1],p,q,x,y)
    57     [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(2)*sqrt(5) - 2/3],
    58      [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(2)*sqrt(5) - 2/3]]
     57    [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(5)*sqrt(2) - 2/3],
     58     [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(5)*sqrt(2) - 2/3]]
    5959
    6060Um eine numerische Approximation der Lösungen zu erhalten können Sie
    6161stattdessen wie folgt vorgehen:
     
    140140
    141141    sage: f = 1/((1+x)*(x-1))
    142142    sage: f.partial_fraction(x)
    143     1/2/(x - 1) - 1/2/(x + 1)
     143    -1/2/(x + 1) + 1/2/(x - 1)
    144144
    145145.. _section-systems:
    146146
     
    173173    sage: t = var("t")
    174174    sage: f = t^2*exp(t) - sin(t)
    175175    sage: f.laplace(t,s)
    176     2/(s - 1)^3 - 1/(s^2 + 1)
     176    -1/(s^2 + 1) + 2/(s - 1)^3
    177177
    178178Hier ist ein komplizierteres Beispiel. Die Verschiebung des
    179179Gleichgewichts einer verkoppelten Feder, die an der linken Wand
  • doc/en/a_tour_of_sage/index.rst

    diff --git a/doc/en/a_tour_of_sage/index.rst b/doc/en/a_tour_of_sage/index.rst
    a b  
    4040
    4141    sage: x = var('x')   # create a symbolic variable
    4242    sage: integrate(sqrt(x)*sqrt(1+x), x)
    43     1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1) - 1/8*log(sqrt(x + 1)/sqrt(x) + 1)
     43    1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x + 1) - 1/8*log(sqrt(x + 1)/sqrt(x) + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1)
    4444
    4545This asks Sage to solve a quadratic equation. The symbol ``==``
    4646represents equality in Sage.
  • doc/en/constructions/calculus.rst

    diff --git a/doc/en/constructions/calculus.rst b/doc/en/constructions/calculus.rst
    a b  
    2525    sage: f = x^3 * e^(k*x) * sin(w*x); f
    2626    x^3*e^(k*x)*sin(w*x)
    2727    sage: f.diff(x)
    28     k*x^3*e^(k*x)*sin(w*x) + w*x^3*e^(k*x)*cos(w*x) + 3*x^2*e^(k*x)*sin(w*x)
     28    w*x^3*cos(w*x)*e^(k*x) + k*x^3*e^(k*x)*sin(w*x) + 3*x^2*e^(k*x)*sin(w*x)
    2929    sage: latex(f.diff(x))
    30     k x^{3} e^{\left(k x\right)} \sin\left(w x\right) + w x^{3} e^{\left(k x\right)} \cos\left(w x\right) + 3 \, x^{2} e^{\left(k x\right)} \sin\left(w x\right)
     30    w x^{3} \cos\left(w x\right) e^{\left(k x\right)} + k x^{3} e^{\left(k x\right)} \sin\left(w x\right) + 3 \, x^{2} e^{\left(k x\right)} \sin\left(w x\right)
    3131
    3232If you type ``view(f.diff(x))`` another window will open up
    3333displaying the compiled output. In the notebook, you can enter
     
    147147    (x, k, w)
    148148    sage: f = x^3 * e^(k*x) * sin(w*x)
    149149    sage: f.integrate(x)
    150     -(((k^6*w + 3*k^4*w^3 + 3*k^2*w^5 + w^7)*x^3 - 24*k^3*w + 24*k*w^3 - 6*(k^5*w + 2*k^3*w^3 + k*w^5)*x^2 + 6*(3*k^4*w + 2*k^2*w^3 - w^5)*x)*e^(k*x)*cos(w*x) - ((k^7 + 3*k^5*w^2 + 3*k^3*w^4 + k*w^6)*x^3 - 6*k^4 + 36*k^2*w^2 - 6*w^4 - 3*(k^6 + k^4*w^2 - k^2*w^4 - w^6)*x^2 + 6*(k^5 - 2*k^3*w^2 - 3*k*w^4)*x)*e^(k*x)*sin(w*x))/(k^8 + 4*k^6*w^2 + 6*k^4*w^4 + 4*k^2*w^6 + w^8)
     150    ((24*k^3*w - 24*k*w^3 - (k^6*w + 3*k^4*w^3 + 3*k^2*w^5 + w^7)*x^3 + 6*(k^5*w + 2*k^3*w^3 + k*w^5)*x^2 - 6*(3*k^4*w + 2*k^2*w^3 - w^5)*x)*cos(w*x)*e^(k*x) - (6*k^4 - 36*k^2*w^2 + 6*w^4 - (k^7 + 3*k^5*w^2 + 3*k^3*w^4 + k*w^6)*x^3 + 3*(k^6 + k^4*w^2 - k^2*w^4 - w^6)*x^2 - 6*(k^5 - 2*k^3*w^2 - 3*k*w^4)*x)*e^(k*x)*sin(w*x))/(k^8 + 4*k^6*w^2 + 6*k^4*w^4 + 4*k^2*w^6 + w^8)
    151151    sage: integrate(1/x^2, x, 1, infinity)
    152152    1
    153153
     
    232232    sage: f2(x) = 1-x
    233233    sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
    234234    sage: f.laplace(x, s)
    235     (s + 1)*e^(-2*s)/s^2 - e^(-s)/s + 1/s - e^(-s)/s^2
     235    -e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2
    236236
    237237For other "reasonable" functions, Laplace transforms can be
    238238computed using the Maxima interface:
     
    283283    sage: y=function('y',x); desolve(diff(y,x,2) + 3*x == y, dvar = y, ics = [1,1,1])
    284284    3*x - 2*e^(x - 1)
    285285    sage: desolve(diff(y,x,2) + 3*x == y, dvar = y)
    286     k1*e^x + k2*e^(-x) + 3*x
     286    k2*e^(-x) + k1*e^x + 3*x
    287287    sage: desolve(diff(y,x) + 3*x == y, dvar = y)
    288288    (3*(x + 1)*e^(-x) + c)*e^x
    289289    sage: desolve(diff(y,x) + 3*x == y, dvar = y, ics = [1,1]).expand()
     
    375375    sage: f.fourier_series_sine_coefficient(2,pi)
    376376    -3/pi
    377377    sage: f.fourier_series_partial_sum(3,pi)
    378     -3*sin(2*x)/pi + sin(x)/pi - 3*cos(x)/pi + 1/4
     378    -3*cos(x)/pi - 3*sin(2*x)/pi + sin(x)/pi + 1/4
    379379
    380380Type ``show(f.plot_fourier_series_partial_sum(15,pi,-5,5))`` and
    381381``show(f.plot_fourier_series_partial_sum_cesaro(15,pi,-5,5))``
  • doc/en/constructions/polynomials.rst

    diff --git a/doc/en/constructions/polynomials.rst b/doc/en/constructions/polynomials.rst
    a b  
    259259    sage: g = f.subs(x = 5/z); g
    260260    (3*y + 25*y/z^2 + 5/z)^3
    261261    sage: h = g.rational_simplify(); h
    262     (27*y^3*z^6 + 135*y^2*z^5 + 225*(3*y^3 + y)*z^4 + 125*(18*y^2 + 1)*z^3 + 
    263     1875*(3*y^3 + y)*z^2 + 15625*y^3 + 9375*y^2*z)/z^6
     262    (27*y^3*z^6 + 135*y^2*z^5 + 225*(3*y^3 + y)*z^4 + 125*(18*y^2 + 1)*z^3 +
     263    15625*y^3 + 9375*y^2*z + 1875*(3*y^3 + y)*z^2)/z^6
    264264
    265265Roots of multivariate polynomials
    266266=================================
  • doc/en/prep/Advanced-2DPlotting.rst

    diff --git a/doc/en/prep/Advanced-2DPlotting.rst b/doc/en/prep/Advanced-2DPlotting.rst
    a b  
    312312    sage: fibonacci
    313313    [(0, 0), (1, 1), (2, 1), (3, 2), (4, 3), (5, 5)]
    314314    sage: asymptotic
    315     [(0, 1/5*sqrt(5)), (1, 1/10*(sqrt(5) + 1)*sqrt(5)), (2, 1/20*(sqrt(5) + 1)^2*sqrt(5)), (3, 1/40*(sqrt(5) + 1)^3*sqrt(5)), (4, 1/80*(sqrt(5) + 1)^4*sqrt(5)), (5, 1/160*(sqrt(5) + 1)^5*sqrt(5))]
     315    [(0, 1/5*sqrt(5)), (1, 1/10*sqrt(5)*(sqrt(5) + 1)), (2, 1/20*sqrt(5)*(sqrt(5) + 1)^2), (3, 1/40*sqrt(5)*(sqrt(5) + 1)^3), (4, 1/80*sqrt(5)*(sqrt(5) + 1)^4), (5, 1/160*sqrt(5)*(sqrt(5) + 1)^5)]
    316316
    317317Now we can plot not just the two sets of points, but also use several of
    318318the documented options for plotting points. Those coming from other
  • doc/en/prep/Calculus.rst

    diff --git a/doc/en/prep/Calculus.rst b/doc/en/prep/Calculus.rst
    a b  
    136136::
    137137
    138138    sage: derivative(sinh(x^2+sqrt(x-1)),x)
    139     1/2*(4*x + 1/sqrt(x - 1))*cosh(sqrt(x - 1) + x^2)
     139    1/2*(4*x + 1/sqrt(x - 1))*cosh(x^2 + sqrt(x - 1))
    140140
    141141And maybe even knows those you don't want.  In this case, we put the
    142142computation inside ``show()`` since the output is so long.
     
    290290
    291291    sage: h(x)=sec(x)
    292292    sage: h.integrate(x)
    293     x |--> log(tan(x) + sec(x))
     293    x |--> log(sec(x) + tan(x))
    294294
    295295Since I defined ``h`` as a function, the answer I get is also a
    296296function.  If I just want an expression as the answer, I can do the
     
    299299::
    300300
    301301    sage: integrate(sec(x),x)
    302     log(tan(x) + sec(x))
     302    log(sec(x) + tan(x))
    303303
    304304Here is another (longer) example.  Do you remember what command would
    305305help it look nicer in the browser?
     
    307307::
    308308
    309309    sage: integrate(1/(1+x^5),x)
    310     1/5*(sqrt(5) - 1)*sqrt(5)*arctan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10))/sqrt(-2*sqrt(5) + 10) + 1/5*(sqrt(5) + 1)*sqrt(5)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10))/sqrt(2*sqrt(5) + 10) - 1/2*(sqrt(5) - 3)*log((sqrt(5) - 1)*x + 2*x^2 + 2)/(5*sqrt(5) - 5) - 1/2*(sqrt(5) + 3)*log(-(sqrt(5) + 1)*x + 2*x^2 + 2)/(5*sqrt(5) + 5) + 1/5*log(x + 1)
     310    1/5*sqrt(5)*(sqrt(5) + 1)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10))/sqrt(2*sqrt(5) + 10) + 1/5*sqrt(5)*(sqrt(5) - 1)*arctan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10))/sqrt(-2*sqrt(5) + 10) - 1/2*(sqrt(5) + 3)*log(2*x^2 - x*(sqrt(5) + 1) + 2)/(5*sqrt(5) + 5) - 1/2*(sqrt(5) - 3)*log(2*x^2 + x*(sqrt(5) - 1) + 2)/(5*sqrt(5) - 5) + 1/5*log(x + 1)
    311311
    312312Some integrals are a little tricky, of course.  If Sage doesn't know the
    313313whole antiderivative, it returns as much of it as it (more properly, as
     
    392392::
    393393
    394394    sage: integral(h,(x,0,pi/8))
    395     -1/2*log(-sin(1/8*pi) + 1) + 1/2*log(sin(1/8*pi) + 1)
     395    1/2*log(sin(1/8*pi) + 1) - 1/2*log(-sin(1/8*pi) + 1)
    396396
    397397Here, just a number might be more helpful.  Sage has several ways of
    398398numerical evaluating integrals.
  • doc/en/thematic_tutorials/tutorial-comprehensions.rst

    diff --git a/doc/en/thematic_tutorials/tutorial-comprehensions.rst b/doc/en/thematic_tutorials/tutorial-comprehensions.rst
    a b  
    322322    x^3 + 2*x^2 + 2*x + 1
    323323
    324324    sage: factor(sum( x^p.length() for p in Permutations(3) ))
    325     (x + 1)*(x^2 + x + 1)
     325    (x^2 + x + 1)*(x + 1)
    326326
    327327    sage: P = Permutations(5)
    328328    sage: all( p in P for p in P )
  • doc/en/tutorial/introduction.rst

    diff --git a/doc/en/tutorial/introduction.rst b/doc/en/tutorial/introduction.rst
    a b  
    4646    1
    4747
    4848    sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k
    49     1/(I*sqrt(3) + 5/9*sqrt(73) + 3/4)
     49    36/(20*sqrt(73) + 36*I*sqrt(3) + 27)
    5050    sage: N(k)
    5151    0.165495678130644 - 0.0521492082074256*I
    5252    sage: N(k,30)      # 30 "bits"
    5353    0.16549568 - 0.052149208*I
    5454    sage: latex(k)
    55     \frac{1}{i \, \sqrt{3} + \frac{5}{9} \, \sqrt{73} + \frac{3}{4}}
     55    \frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}
    5656
    5757.. _installation:
    5858
  • doc/en/tutorial/tour_algebra.rst

    diff --git a/doc/en/tutorial/tour_algebra.rst b/doc/en/tutorial/tour_algebra.rst
    a b  
    5252    sage: eq2 = q*y+p*x==-6
    5353    sage: eq3 = q*y^2+p*x^2==24
    5454    sage: solve([eq1,eq2,eq3,p==1],p,q,x,y)
    55     [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(2)*sqrt(5) - 2/3],
    56      [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(2)*sqrt(5) - 2/3]]
     55    [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(5)*sqrt(2) - 2/3],
     56     [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(5)*sqrt(2) - 2/3]]
    5757
    5858For numerical approximations of the solutions, you can instead use:
    5959
     
    138138
    139139    sage: f = 1/((1+x)*(x-1))
    140140    sage: f.partial_fraction(x)
    141     1/2/(x - 1) - 1/2/(x + 1)
     141    -1/2/(x + 1) + 1/2/(x - 1)
    142142
    143143.. _section-systems:
    144144
     
    170170    sage: t = var("t")
    171171    sage: f = t^2*exp(t) - sin(t)
    172172    sage: f.laplace(t,s)
    173     2/(s - 1)^3 - 1/(s^2 + 1)
     173    -1/(s^2 + 1) + 2/(s - 1)^3
    174174
    175175Here is a more involved example. The displacement from equilibrium
    176176(respectively) for a coupled spring attached to a wall on the left
  • doc/fr/a_tour_of_sage/index.rst

    diff --git a/doc/fr/a_tour_of_sage/index.rst b/doc/fr/a_tour_of_sage/index.rst
    a b  
    4343
    4444    sage: x = var('x')   # Créer une variable symbolique
    4545    sage: integrate(sqrt(x)*sqrt(1+x), x)
    46     1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1) - 1/8*log(sqrt(x + 1)/sqrt(x) + 1)
     46    1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x + 1) - 1/8*log(sqrt(x + 1)/sqrt(x) + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1)
    4747
    4848Ceci permet de demander à Sage de résoudre une équation
    4949quadratique. Le symbole ``==`` représente l'égalité sous Sage.
     
    140140================================
    141141
    142142Quand vous utilisez Sage, vous avez accès à l'une des plus grandes
    143 collections Open Source d'algorithmes de calcul.
    144  No newline at end of file
     143collections Open Source d'algorithmes de calcul.
  • doc/fr/tutorial/introduction.rst

    diff --git a/doc/fr/tutorial/introduction.rst b/doc/fr/tutorial/introduction.rst
    a b  
    4747    1
    4848   
    4949    sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k
    50     1/(I*sqrt(3) + 5/9*sqrt(73) + 3/4)
     50    36/(20*sqrt(73) + 36*I*sqrt(3) + 27)
    5151    sage: N(k)
    5252    0.165495678130644 - 0.0521492082074256*I
    5353    sage: N(k,30)      # 30 "bits"
    5454    0.16549568 - 0.052149208*I
    5555    sage: latex(k)
    56     \frac{1}{i \, \sqrt{3} + \frac{5}{9} \, \sqrt{73} + \frac{3}{4}}
     56    \frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}
    5757
    5858.. _installation:
    5959
  • doc/fr/tutorial/tour_algebra.rst

    diff --git a/doc/fr/tutorial/tour_algebra.rst b/doc/fr/tutorial/tour_algebra.rst
    a b  
    5050    sage: eq2 = q*y+p*x==-6
    5151    sage: eq3 = q*y^2+p*x^2==24
    5252    sage: solve([eq1,eq2,eq3,p==1],p,q,x,y)
    53     [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(2)*sqrt(5) - 2/3],
    54      [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(2)*sqrt(5) - 2/3]]
     53    [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(5)*sqrt(2) - 2/3], [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(5)*sqrt(2) - 2/3]]
    5554
    5655Pour une résolution numérique, on peut utiliser à la place :
    5756
     
    117116
    118117    sage: f = 1/((1+x)*(x-1))
    119118    sage: f.partial_fraction(x)
    120     1/2/(x - 1) - 1/2/(x + 1)
     119    -1/2/(x + 1) + 1/2/(x - 1)
    121120
    122121.. _section-systems:
    123122
     
    150149    sage: t = var("t")
    151150    sage: f = t^2*exp(t) - sin(t)
    152151    sage: f.laplace(t,s)
    153     2/(s - 1)^3 - 1/(s^2 + 1)
     152    -1/(s^2 + 1) + 2/(s - 1)^3
    154153
    155154Voici un exemple plus élaboré. L'élongation à partir du point
    156155d'équilibre de ressorts couplés attachés à gauche à un mur
  • doc/ru/tutorial/introduction.rst

    diff --git a/doc/ru/tutorial/introduction.rst b/doc/ru/tutorial/introduction.rst
    a b  
    4747    1
    4848
    4949    sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k
    50     1/(I*sqrt(3) + 5/9*sqrt(73) + 3/4)
     50    36/(20*sqrt(73) + 36*I*sqrt(3) + 27)
    5151    sage: N(k)
    5252    0.165495678130644 - 0.0521492082074256*I
    5353    sage: N(k,30)      # Точность 30 бит
    5454    0.16549568 - 0.052149208*I
    5555    sage: latex(k)
    56     \frac{1}{i \, \sqrt{3} + \frac{5}{9} \, \sqrt{73} + \frac{3}{4}}
     56    \frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}
    5757
    5858.. _installation:
    5959
  • doc/ru/tutorial/tour_algebra.rst

    diff --git a/doc/ru/tutorial/tour_algebra.rst b/doc/ru/tutorial/tour_algebra.rst
    a b  
    4949    sage: eq2 = q*y+p*x==-6
    5050    sage: eq3 = q*y^2+p*x^2==24
    5151    sage: solve([eq1,eq2,eq3,p==1],p,q,x,y)
    52     [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(2)*sqrt(5) - 2/3],
    53      [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(2)*sqrt(5) - 2/3]]
     52    [[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(5)*sqrt(2) - 2/3], [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(5)*sqrt(2) - 2/3]]
    5453
    5554Для приближенных значений решения можно использовать:
    5655
     
    135134
    136135    sage: f = 1/((1+x)*(x-1))
    137136    sage: f.partial_fraction(x)
    138     1/2/(x - 1) - 1/2/(x + 1)
     137    -1/2/(x + 1) + 1/2/(x - 1)
    139138
    140139.. _section-systems:
    141140
     
    167166    sage: t = var("t")
    168167    sage: f = t^2*exp(t) - sin(t)
    169168    sage: f.laplace(t,s)
    170     2/(s - 1)^3 - 1/(s^2 + 1)
     169    -1/(s^2 + 1) + 2/(s - 1)^3
    171170
    172171Приведем более сложный пример. Отклонение от положения равновесия для пары
    173172пружин, прикрепленных к стене слева,
  • doc/tr/a_tour_of_sage/index.rst

    diff --git a/doc/tr/a_tour_of_sage/index.rst b/doc/tr/a_tour_of_sage/index.rst
    a b  
    3737
    3838    sage: x = var('x')   # değişkeni sembolik olarak yaratıyoruz
    3939    sage: integrate(sqrt(x)*sqrt(1+x), x)
    40     1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1) - 1/8*log(sqrt(x + 1)/sqrt(x) + 1)
     40    1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x + 1) - 1/8*log(sqrt(x + 1)/sqrt(x) + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1)
    4141
    4242Bu komut Sage'e ikinci derece denklemi çözdürür. ``==`` sembolü Sage'de eşitlik anlamına gelir.
    4343
     
    126126Sage'de Algoritmaların Kullanımı
    127127================================
    128128
    129 Sage kullanırken dünyanın en geniş açık kaynak hesaplama algoritma koleksiyonlarından biriyle çalışırsınız.
    130  No newline at end of file
     129Sage kullanırken dünyanın en geniş açık kaynak hesaplama algoritma koleksiyonlarından biriyle çalışırsınız.
  • sage/calculus/calculus.py

    diff --git a/sage/calculus/calculus.py b/sage/calculus/calculus.py
    a b  
    109109    [   (r, theta) |--> cos(theta) (r, theta) |--> -r*sin(theta)]
    110110    [   (r, theta) |--> sin(theta)  (r, theta) |--> r*cos(theta)]
    111111    sage: T.diff().det() # Jacobian
    112     (r, theta) |--> r*sin(theta)^2 + r*cos(theta)^2
     112    (r, theta) |--> r*cos(theta)^2 + r*sin(theta)^2
    113113
    114114When the order of variables is ambiguous, Sage will raise an
    115115exception when differentiating::
     
    206206::
    207207
    208208    sage: f(x,y) = log(x)*cos(y); f
    209     (x, y) |--> log(x)*cos(y)
     209    (x, y) |--> cos(y)*log(x)
    210210
    211211Then we have fixed an order of variables and there is no ambiguity
    212212substituting or evaluating::
     
    269269    sage: ComplexField(200)(sin(I))
    270270    1.1752011936438014568823818505956008151557179813340958702296*I
    271271    sage: f = sin(I) + cos(I/2); f
    272     sin(I) + cos(1/2*I)
     272    cos(1/2*I) + sin(I)
    273273    sage: CC(f)
    274274    1.12762596520638 + 1.17520119364380*I
    275275    sage: ComplexField(200)(f)
     
    370370Check that the problem with Taylor expansions of the gamma function
    371371(Trac #9217) is fixed::
    372372
    373     sage: taylor(gamma(1/3+x),x,0,3)      # random output - remove this in trac #9880
    374     -1/432*((36*(pi*sqrt(3) + 9*log(3))*euler_gamma^2 + 27*pi^2*log(3) + 72*euler_gamma^3 + 243*log(3)^3 + 18*(6*pi*sqrt(3)*log(3) + pi^2 + 27*log(3)^2 + 12*psi(1, 1/3))*euler_gamma + 324*psi(1, 1/3)*log(3) + (pi^3 + 9*(9*log(3)^2 + 4*psi(1, 1/3))*pi)*sqrt(3))*gamma(1/3) - 72*gamma(1/3)*psi(2, 1/3))*x^3 + 1/24*(6*pi*sqrt(3)*log(3) + 4*(pi*sqrt(3) + 9*log(3))*euler_gamma + pi^2 + 12*euler_gamma^2 + 27*log(3)^2 + 12*psi(1, 1/3))*x^2*gamma(1/3) - 1/6*(6*euler_gamma + pi*sqrt(3) + 9*log(3))*x*gamma(1/3) + gamma(1/3)
     373    sage: taylor(gamma(1/3+x),x,0,3)
     374    -1/432*((72*euler_gamma^3 + 36*euler_gamma^2*(sqrt(3)*pi + 9*log(3)) +
     375    27*pi^2*log(3) + 243*log(3)^3 + 18*euler_gamma*(6*sqrt(3)*pi*log(3) + pi^2
     376    + 27*log(3)^2 + 12*psi(1, 1/3)) + 324*log(3)*psi(1, 1/3) + sqrt(3)*(pi^3 +
     377    9*pi*(9*log(3)^2 + 4*psi(1, 1/3))))*gamma(1/3) - 72*psi(2,
     378    1/3)*gamma(1/3))*x^3 + 1/24*(6*sqrt(3)*pi*log(3) + 12*euler_gamma^2 + pi^2
     379    + 4*euler_gamma*(sqrt(3)*pi + 9*log(3)) + 27*log(3)^2 + 12*psi(1,
     380    1/3))*x^2*gamma(1/3) - 1/6*(6*euler_gamma + sqrt(3)*pi +
     381    9*log(3))*x*gamma(1/3) + gamma(1/3)
    375382    sage: map(lambda f:f[0].n(), _.coeffs())  # numerical coefficients to make comparison easier; Maple 12 gives same answer
    376383    [2.6789385347..., -8.3905259853..., 26.662447494..., -80.683148377...]
    377384
     
    379386
    380387    sage: k = var("k")
    381388    sage: sum(1/(1+k^2), k, -oo, oo)
    382     1/2*I*psi(-I) - 1/2*I*psi(I) + 1/2*I*psi(-I + 1) - 1/2*I*psi(I + 1)
     389    -1/2*I*psi(I + 1) + 1/2*I*psi(-I + 1) - 1/2*I*psi(I) + 1/2*I*psi(-I)
    383390
    384391Ensure that ticket #8624 is fixed::
    385392
     
    487494    ::
    488495
    489496        sage: symbolic_sum(k * binomial(n, k), k, 1, n)
    490         n*2^(n - 1)
     497        2^(n - 1)*n
    491498
    492499    ::
    493500
     
    865872
    866873        sage: f = x^3 - x + 1
    867874        sage: a = f.solve(x)[0].rhs(); a
    868         -1/2*(I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(23) - 1/2)^(1/3) - 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(3)*sqrt(23) - 1/2)^(1/3)
     875        -1/2*(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)
    869876        sage: a.minpoly()
    870877        x^3 - x + 1
    871878
     
    878885        sage: f = a.minpoly(); f
    879886        x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576
    880887        sage: f(a)
    881         ((((sqrt(2) + sqrt(3) + sqrt(5))^2 - 40)*(sqrt(2) + sqrt(3) + sqrt(5))^2 + 352)*(sqrt(2) + sqrt(3) + sqrt(5))^2 - 960)*(sqrt(2) + sqrt(3) + sqrt(5))^2 + 576
     888        ((((sqrt(5) + sqrt(3) + sqrt(2))^2 - 40)*(sqrt(5) + sqrt(3) + sqrt(2))^2 + 352)*(sqrt(5) + sqrt(3) + sqrt(2))^2 - 960)*(sqrt(5) + sqrt(3) + sqrt(2))^2 + 576
    882889        sage: f(a).expand()
    883890        0
    884891
     
    13011308        sage: xt = E[0,2].inverse_laplace(s,t)
    13021309        sage: yt = E[1,2].inverse_laplace(s,t)
    13031310        sage: xt
    1304         629/2*e^(-4*t) - 91/2*e^(4*t) + 1
     1311        -91/2*e^(4*t) + 629/2*e^(-4*t) + 1
    13051312        sage: yt
    1306         629/8*e^(-4*t) + 91/8*e^(4*t)
     1313        91/8*e^(4*t) + 629/8*e^(-4*t)
    13071314        sage: p1 = plot(xt,0,1/2,rgbcolor=(1,0,0))
    13081315        sage: p2 = plot(yt,0,1/2,rgbcolor=(0,1,0))
    13091316        sage: (p1+p2).save(os.path.join(SAGE_TMP, "de_plot.png"))
     
    17261733    Trac #8459 fixed::
    17271734
    17281735        sage: maxima('3*li[2](u)+8*li[33](exp(u))').sage()
    1729         3*polylog(2, u) + 8*polylog(33, e^u)
     1736        8*polylog(33, e^u) + 3*polylog(2, u)
    17301737
    17311738    Check if #8345 is fixed::
    17321739
  • sage/calculus/desolvers.py

    diff --git a/sage/calculus/desolvers.py b/sage/calculus/desolvers.py
    a b  
    140140        sage: y = function('y', x)
    141141        sage: de = diff(y,x,2) - y == x
    142142        sage: desolve(de, y)
    143         k1*e^x + k2*e^(-x) - x
     143        k2*e^(-x) + k1*e^x - x
    144144       
    145145
    146146    ::
    147147
    148148        sage: f = desolve(de, y, [10,2,1]); f
    149         -x + 5*e^(-x + 10) + 7*e^(x - 10)
     149        -x + 7*e^(x - 10) + 5*e^(-x + 10)
    150150
    151151    ::
    152152
     
    162162
    163163        sage: de = diff(y,x,2) + y == 0
    164164        sage: desolve(de, y)
    165         k1*sin(x) + k2*cos(x)
     165        k2*cos(x) + k1*sin(x)
    166166
    167167    ::
    168168
    169169        sage: desolve(de, y, [0,1,pi/2,4])
    170         4*sin(x) + cos(x)
     170        cos(x) + 4*sin(x)
    171171
    172172    ::
    173173
     
    219219    ::
    220220
    221221        sage: desolve(diff(y,x)*sin(y) == cos(x),y,[pi/2,1])
    222         -cos(y(x)) == sin(x) - cos(1) - 1
     222        -cos(y(x)) == -cos(1) + sin(x) - 1
    223223
    224224    Linear equation - Sage returns the expression on the right hand side only::
    225225
    226226        sage: desolve(diff(y,x)+(y) == cos(x),y)
    227         1/2*((sin(x) + cos(x))*e^x + 2*c)*e^(-x)
     227        1/2*((cos(x) + sin(x))*e^x + 2*c)*e^(-x)
    228228
    229229    ::
    230230
    231231        sage: desolve(diff(y,x)+(y) == cos(x),y,show_method=True)
    232         [1/2*((sin(x) + cos(x))*e^x + 2*c)*e^(-x), 'linear']
     232        [1/2*((cos(x) + sin(x))*e^x + 2*c)*e^(-x), 'linear']
    233233
    234234    ::
    235235
    236236        sage: desolve(diff(y,x)+(y) == cos(x),y,[0,1])
    237         1/2*(e^x*sin(x) + e^x*cos(x) + 1)*e^(-x)
     237        1/2*(cos(x)*e^x + e^x*sin(x) + 1)*e^(-x)
    238238
    239239    This ODE with separated variables is solved as
    240240    exact. Explanation - factor does not split `e^{x-y}` in Maxima
     
    312312    ::
    313313       
    314314        sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2])
    315         3*((e^(1/2*pi) - 2)*x/pi + 1)*e^(-x) + 1/2*sin(x)
     315        3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x)
    316316       
    317317    ::
    318318       
    319319        sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2],show_method=True)
    320         [3*((e^(1/2*pi) - 2)*x/pi + 1)*e^(-x) + 1/2*sin(x), 'variationofparameters']
     320        [3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x), 'variationofparameters']
    321321       
    322322    ::
    323323       
     
    342342    ::
    343343       
    344344        sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2])
    345         (2*(2*e^(1/2*pi) - 3)*x/pi + 3)*e^(-x)
     345        (2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x)
    346346       
    347347    ::
    348348       
    349349        sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2],show_method=True)
    350         [(2*(2*e^(1/2*pi) - 3)*x/pi + 3)*e^(-x), 'constcoeff']
     350        [(2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x), 'constcoeff']
    351351       
    352352    TESTS:
    353353   
     
    365365        sage: sage.calculus.calculus.maxima('domain:complex')  # back to the default complex domain
    366366        complex
    367367        sage: desolve(x*diff(y,x)-x*sqrt(y^2+x^2)-y == 0, y, contrib_ode=True)
    368         [1/2*(2*x^2*sqrt(x^(-2)) - 2*x*sqrt(x^(-2))*arcsinh(y(x)/sqrt(x^2))
    369         - 2*x*sqrt(x^(-2))*arcsinh(y(x)^2/(sqrt(y(x)^2)*x))
    370         + log(4*(2*x^2*sqrt((x^2*y(x)^2 + y(x)^4)/x^2)*sqrt(x^(-2)) + x^2 + 2*y(x)^2)/x^2))/(x*sqrt(x^(-2))) == c]
     368        [1/2*(2*x^2*sqrt(x^(-2)) - 2*x*sqrt(x^(-2))*arcsinh(y(x)/sqrt(x^2)) -
     369            2*x*sqrt(x^(-2))*arcsinh(y(x)^2/(x*sqrt(y(x)^2))) +
     370            log(4*(2*x^2*sqrt((x^2*y(x)^2 + y(x)^4)/x^2)*sqrt(x^(-2)) + x^2 +
     371            2*y(x)^2)/x^2))/(x*sqrt(x^(-2))) == c]
    371372
    372373    Trac #6479 fixed::
    373374
     
    702703        sage: de2 = diff(y,t) - x + 1 == 0
    703704        sage: desolve_system([de1, de2], [x,y])
    704705        [x(t) == (x(0) - 1)*cos(t) - (y(0) - 1)*sin(t) + 1,
    705          y(t) == (x(0) - 1)*sin(t) + (y(0) - 1)*cos(t) + 1]
     706         y(t) == (y(0) - 1)*cos(t) + (x(0) - 1)*sin(t) + 1]
    706707         
    707708    Now we give some initial conditions::
    708709   
  • sage/calculus/functional.py

    diff --git a/sage/calculus/functional.py b/sage/calculus/functional.py
    a b  
    9393        (a, x)
    9494        sage: f = exp(sin(a - x^2))/x
    9595        sage: derivative(f, x)
    96         -2*e^(sin(-x^2 + a))*cos(-x^2 + a) - e^(sin(-x^2 + a))/x^2
     96        -2*cos(-x^2 + a)*e^(sin(-x^2 + a)) - e^(sin(-x^2 + a))/x^2
    9797        sage: derivative(f, a)
    98         e^(sin(-x^2 + a))*cos(-x^2 + a)/x
     98        cos(-x^2 + a)*e^(sin(-x^2 + a))/x
    9999   
    100100    Syntax for repeated differentiation::
    101101   
     
    167167    ::
    168168   
    169169        sage: integral(x/(x^3-1), x)
    170         1/3*sqrt(3)*arctan(1/3*(2*x + 1)*sqrt(3)) + 1/3*log(x - 1) - 1/6*log(x^2 + x + 1)
     170        1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*log(x^2 + x + 1) + 1/3*log(x - 1)
    171171   
    172172    ::
    173173   
     
    214214    Note, however, that::
    215215   
    216216        sage: integral( exp(-x^2)*ln(x), x, 0, oo)
    217         -1/4*(euler_gamma + 2*log(2))*sqrt(pi)
     217        -1/4*sqrt(pi)*(euler_gamma + 2*log(2))
    218218   
    219219    This definite integral is easy::
    220220   
     
    371371    Taylor polynomial in two variables::
    372372
    373373        sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,-1),4)
    374         (y + 1)^3*(x - 1) + (y + 1)^3 - 3*(y + 1)^2*(x - 1) - 3*(y + 1)^2 + 3*(y + 1)*(x - 1) - x + 3*y + 3
     374        (x - 1)*(y + 1)^3 - 3*(x - 1)*(y + 1)^2 + (y + 1)^3 + 3*(x - 1)*(y + 1) - 3*(y + 1)^2 - x + 3*y + 3
    375375    """
    376376    if not isinstance(f, Expression):
    377377        f = SR(f)
     
    382382    EXAMPLES::
    383383   
    384384        sage: a = (x-1)*(x^2 - 1); a
    385         (x - 1)*(x^2 - 1)
     385        (x^2 - 1)*(x - 1)
    386386        sage: expand(a)
    387387        x^3 - x^2 - x + 1
    388388   
  • sage/calculus/functions.py

    diff --git a/sage/calculus/functions.py b/sage/calculus/functions.py
    a b  
    5454    Two-by-two Wronskian of sin(x) and e^x::
    5555   
    5656        sage: wronskian(sin(x), e^x, x)
    57         e^x*sin(x) - e^x*cos(x)
     57        -cos(x)*e^x + e^x*sin(x)
    5858
    5959    Or don't put x last::
    6060   
    6161        sage: wronskian(x, sin(x), e^x)
    62         (e^x*sin(x) + e^x*cos(x))*x - 2*e^x*sin(x)
     62        (cos(x)*e^x + e^x*sin(x))*x - 2*e^x*sin(x)
    6363
    6464    Example where one of the functions is constant::
    6565   
  • sage/calculus/test_sympy.py

    diff --git a/sage/calculus/test_sympy.py b/sage/calculus/test_sympy.py
    a b  
    164164    sage: type(e)
    165165    <type 'sage.symbolic.expression.Expression'>
    166166    sage: e
    167     sin(y) + cos(x)
     167    cos(x) + sin(y)
    168168    sage: e = sage.all.cos(var("y")**3)**4+var("x")**2
    169169    sage: e = e._sympy_()
    170170    sage: e
  • sage/calculus/tests.py

    diff --git a/sage/calculus/tests.py b/sage/calculus/tests.py
    a b  
    2626::
    2727
    2828    sage: christoffel(3,3,2, [t,r,theta,phi], m)
    29     -sin(theta)*cos(theta)
     29    -cos(theta)*sin(theta)
    3030    sage: X = christoffel(1,1,1,[t,r,theta,phi],m)
    3131    sage: X
    32      1/2/((1/r - 1)*r^2)
     32    1/2/(r^2*(1/r - 1))
    3333    sage: X.rational_simplify()
    3434     -1/2/(r^2 - r)
    3535
     
    6868    sage: g(x) = cos(x) + x^3
    6969    sage: u = f(x+t) + g(x-t)
    7070    sage: u
    71     -(t - x)^3 + sin((t + x)^2) + cos(-t + x)
     71    -(t - x)^3 + cos(-t + x) + sin((t + x)^2)
    7272    sage: u.diff(t,2) - u.diff(x,2)
    7373    0
    7474
     
    9595    sage: derivative(arctan(x), x)
    9696    1/(x^2 + 1)
    9797    sage: derivative(x^n, x, 3)
    98     (n - 2)*(n - 1)*n*x^(n - 3)
     98    (n - 1)*(n - 2)*n*x^(n - 3)
    9999    sage: derivative( function('f')(x), x)
    100100    D[0](f)(x)   
    101101    sage: diff( 2*x*f(x^2), x)
    102102    4*x^2*D[0](f)(x^2) + 2*f(x^2)
    103103    sage: integrate( 1/(x^4 - a^4), x)
    104     1/4*log(-a + x)/a^3 - 1/4*log(a + x)/a^3 - 1/2*arctan(x/a)/a^3
     104    -1/2*arctan(x/a)/a^3 - 1/4*log(a + x)/a^3 + 1/4*log(-a + x)/a^3
    105105    sage: expand(integrate(log(1-x^2), x))
    106     x*log(-x^2 + 1) - 2*x - log(x - 1) + log(x + 1)
     106    x*log(-x^2 + 1) - 2*x + log(x + 1) - log(x - 1)
    107107    sage: integrate(log(1-x^2)/x, x)
    108     1/2*log(-x^2 + 1)*log(x^2) + 1/2*polylog(2, -x^2 + 1)
     108    1/2*log(x^2)*log(-x^2 + 1) + 1/2*polylog(2, -x^2 + 1)
    109109    sage: integrate(exp(1-x^2),x)
    110     1/2*sqrt(pi)*e*erf(x)
     110    1/2*sqrt(pi)*erf(x)*e
    111111    sage: integrate(sin(x^2),x)
    112     1/8*((I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x) + (I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x))*sqrt(pi)
     112    1/8*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x))
    113113
    114114    sage: integrate((1-x^2)^n,x)
    115115    integrate((-x^2 + 1)^n, x)
    116116    sage: integrate(x^x,x)
    117117    integrate(x^x, x)
    118118    sage: integrate(1/(x^3+1),x)
    119     1/3*sqrt(3)*arctan(1/3*(2*x - 1)*sqrt(3)) + 1/3*log(x + 1) - 1/6*log(x^2 - x + 1)
     119    1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/6*log(x^2 - x + 1) + 1/3*log(x + 1)
    120120    sage: integrate(1/(x^3+1), x, 0, 1)
    121     1/9*pi*sqrt(3) + 1/3*log(2)
     121    1/9*sqrt(3)*pi + 1/3*log(2)
    122122
    123123::
    124124
     
    159159    sage: diff(sin(x), x, 3)
    160160    -cos(x)
    161161    sage: diff(x*sin(cos(x)), x)
    162     -x*sin(x)*cos(cos(x)) + sin(cos(x))
     162    -x*cos(cos(x))*sin(x) + sin(cos(x))
    163163    sage: diff(tan(x), x)
    164164    tan(x)^2 + 1
    165165    sage: f = function('f'); f
     
    190190::
    191191
    192192    sage: integrate( x/(x^3-1), x)
    193     1/3*sqrt(3)*arctan(1/3*(2*x + 1)*sqrt(3)) + 1/3*log(x - 1) - 1/6*log(x^2 + x + 1)
     193    1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*log(x^2 + x + 1) + 1/3*log(x - 1)
    194194    sage: integrate(exp(-x^2), x)
    195195    1/2*sqrt(pi)*erf(x)   
    196196    sage: integrate(exp(-x^2)*log(x), x)       # todo: maple can compute this exactly.
     
    208208    sage: function('f, g')
    209209    (f, g)
    210210    sage: diff(f(t)*g(t),t)
    211     f(t)*D[0](g)(t) + g(t)*D[0](f)(t)
     211    g(t)*D[0](f)(t) + f(t)*D[0](g)(t)
    212212    sage: diff(f(t)/g(t), t)
    213     -f(t)*D[0](g)(t)/g(t)^2 + D[0](f)(t)/g(t)
     213    D[0](f)(t)/g(t) - f(t)*D[0](g)(t)/g(t)^2
    214214    sage: diff(f(t) + g(t), t)
    215215    D[0](f)(t) + D[0](g)(t)   
    216216    sage: diff(c*f(t), t)
  • sage/calculus/var.pyx

    diff --git a/sage/calculus/var.pyx b/sage/calculus/var.pyx
    a b  
    220220    method to replace functions::
    221221   
    222222        sage: k.substitute_function(supersin, sin)
    223         2*sin(x)*cos(x)
     223        2*cos(x)*sin(x)
    224224    """
    225225    if len(args) > 0:
    226226        return function(s, **kwds)(*args)
     
    254254        ...
    255255        NameError: name 'x' is not defined
    256256        sage: expand((e + i)^2)
    257         2*I*e + e^2 - 1
     257        e^2 + 2*I*e - 1
    258258        sage: k
    259259        15       
    260260    """
  • sage/calculus/wester.py

    diff --git a/sage/calculus/wester.py b/sage/calculus/wester.py
    a b  
    2929
    3030    sage: # (YES) Evaluate  e^(Pi*Sqrt(163)) to 50 decimal digits
    3131    sage: a = e^(pi*sqrt(163)); a
    32     e^(pi*sqrt(163))
     32    e^(sqrt(163)*pi)
    3333    sage: print RealField(150)(a)
    3434    2.6253741264076874399999999999925007259719820e17
    3535
     
    156156
    157157    sage: # (YES) Factorize x^100-1.
    158158    sage: factor(x^100-1)
    159     (x - 1)*(x + 1)*(x^2 + 1)*(x^4 - x^3 + x^2 - x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^6 + x^4 - x^2 + 1)*(x^20 - x^15 + x^10 - x^5 + 1)*(x^20 + x^15 + x^10 + x^5 + 1)*(x^40 - x^30 + x^20 - x^10 + 1)
     159    (x^40 - x^30 + x^20 - x^10 + 1)*(x^20 + x^15 + x^10 + x^5 + 1)*(x^20 - x^15 + x^10 - x^5 + 1)*(x^8 - x^6 + x^4 - x^2 + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1)*(x^2 + 1)*(x + 1)*(x - 1)
    160160    sage: # Also, algebraically
    161161    sage: x = polygen(QQ)
    162162    sage: factor(x^100 - 1)
     
    193193    sage: f = (x^2+2*x+3)/(x^3+4*x^2+5*x+2); f
    194194    (x^2 + 2*x + 3)/(x^3 + 4*x^2 + 5*x + 2)
    195195    sage: f.partial_fraction()
    196     -2/(x + 1) + 2/(x + 1)^2 + 3/(x + 2)
     196    3/(x + 2) - 2/(x + 1) + 2/(x + 1)^2
    197197
    198198::
    199199
     
    233233    sage: eqn = prod(x-i for i in range(1,5 +1)) < 0
    234234    sage: # but don't know how to solve
    235235    sage: eqn
    236     (x - 5)*(x - 4)*(x - 3)*(x - 2)*(x - 1) < 0
     236    (x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) < 0
    237237
    238238::
    239239
     
    287287
    288288    sage: # (YES) (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3))-6 = 0
    289289    sage: a = (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6; a
    290     (2^(1/3) + 4^(1/3))^3 - 6*2^(1/3) - 6*4^(1/3) - 6
     290    (4^(1/3) + 2^(1/3))^3 - 6*4^(1/3) - 6*2^(1/3) - 6
    291291    sage: bool(a==0)
    292292    True
    293293    sage: abs(float(a)) < 1e-10
     
    339339    sage: assume(r>0)
    340340    sage: f = (4*r+4*sqrt(r)+1)^(sqrt(r)/(2*sqrt(r)+1))*(2*sqrt(r)+1)^(2*sqrt(r)+1)^(-1)-2*sqrt(r)-1
    341341    sage: f
    342     (2*sqrt(r) + 1)^(1/(2*sqrt(r) + 1))*(4*r + 4*sqrt(r) + 1)^(sqrt(r)/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1
     342    (4*r + 4*sqrt(r) + 1)^(sqrt(r)/(2*sqrt(r) + 1))*(2*sqrt(r) + 1)^(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1
    343343    sage: bool(f == 0)
    344344    False
    345345    sage: [abs(float(f(r=i))) < 1e-10 for i in [0.1,0.3,0.5]]
     
    362362    sage: a = tan(z); a
    363363    tan(z)
    364364    sage: a.real()
    365     tan(real_part(z))/(tan(real_part(z))^2*tan(imag_part(z))^2 + 1)
     365    tan(real_part(z))/(tan(imag_part(z))^2*tan(real_part(z))^2 + 1)
    366366    sage: a.imag()
    367     tanh(imag_part(z))/(tan(real_part(z))^2*tan(imag_part(z))^2 + 1)
     367    tanh(imag_part(z))/(tan(imag_part(z))^2*tan(real_part(z))^2 + 1)
    368368
    369369
    370370::
     
    455455    [  1   d d^2 d^3]
    456456    sage: d = m.determinant()
    457457    sage: d.factor()
    458     (c - d)*(b - d)*(b - c)*(a - d)*(a - c)*(a - b)
     458    (a - b)*(a - c)*(a - d)*(b - c)*(b - d)*(c - d)
    459459
    460460::
    461461
     
    555555    sage: # (YES) Taylor expansion of Ln(x)^a*Exp(-b*x) at x=1.
    556556    sage: a,b = var('a,b')
    557557    sage: taylor(log(x)^a*exp(-b*x), x, 1, 3)
    558     -1/48*(x - 1)^3*((6*b + 5)*(x - 1)^a*a^2 + (x - 1)^a*a^3 + 8*(x - 1)^a*b^3 + 2*(6*b^2 + 5*b + 3)*(x - 1)^a*a)*e^(-b) + 1/24*(x - 1)^2*((12*b + 5)*(x - 1)^a*a + 3*(x - 1)^a*a^2 + 12*(x - 1)^a*b^2)*e^(-b) - 1/2*(x - 1)*((x - 1)^a*a + 2*(x - 1)^a*b)*e^(-b) + (x - 1)^a*e^(-b)
     558    -1/48*(a^3*(x - 1)^a + a^2*(6*b + 5)*(x - 1)^a + 8*b^3*(x - 1)^a + 2*(6*b^2 + 5*b + 3)*a*(x - 1)^a)*(x - 1)^3*e^(-b) + 1/24*(3*a^2*(x - 1)^a + a*(12*b + 5)*(x - 1)^a + 12*b^2*(x - 1)^a)*(x - 1)^2*e^(-b) - 1/2*(a*(x - 1)^a + 2*b*(x - 1)^a)*(x - 1)*e^(-b) + (x - 1)^a*e^(-b)
    559559   
    560560::
    561561
  • sage/categories/classical_crystals.py

    diff --git a/sage/categories/classical_crystals.py b/sage/categories/classical_crystals.py
    a b  
    156156                sage: weight.reduced_word()
    157157                [2, 1]
    158158                sage: T.demazure_character(weight)
    159                 x1^2*x2 + x1^2*x3 + x1*x2^2 + x1*x2*x3 + x1*x3^2
     159                x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x1*x3^2
    160160
    161161                sage: T = CrystalOfTableaux(['A',3],shape=[2,1])
    162162                sage: T.demazure_character([1,2,3])
    163                 x1^2*x2 + x1^2*x3 + x1*x2^2 + x1*x2*x3 + x2^2*x3
     163                x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3
    164164                sage: W = WeylGroup(['A',3])
    165165                sage: w = W.from_reduced_word([1,2,3])
    166166                sage: T.demazure_character(w)
    167                 x1^2*x2 + x1^2*x3 + x1*x2^2 + x1*x2*x3 + x2^2*x3
     167                x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3
    168168
    169169                sage: T = CrystalOfTableaux(['B',2], shape = [2])
    170170                sage: e = T.weight_lattice_realization().basis()
  • sage/coding/code_bounds.py

    diff --git a/sage/coding/code_bounds.py b/sage/coding/code_bounds.py
    a b  
    451451        sage: entropy(0, 2)
    452452        0
    453453        sage: entropy(1/5,4)
    454         -1/5*log(1/5)/log(4) - 4/5*log(4/5)/log(4) + 1/5*log(3)/log(4)
     454        1/5*log(3)/log(4) - 4/5*log(4/5)/log(4) - 1/5*log(1/5)/log(4)
    455455        sage: entropy(1, 3)
    456456        log(2)/log(3)
    457457
  • sage/combinat/partition.py

    diff --git a/sage/combinat/partition.py b/sage/combinat/partition.py
    a b  
    20882088        EXAMPLES::
    20892089
    20902090            sage: Partition([3,2,1]).hook_product(x)
    2091             (x + 2)^2*(2*x + 3)
     2091            (2*x + 3)*(x + 2)^2
    20922092            sage: Partition([2,2]).hook_product(x)
    2093             2*(x + 1)*(x + 2)
     2093            2*(x + 2)*(x + 1)
    20942094        """
    20952095
    20962096        nu = self.conjugate()
     
    36663666            abs(x)
    36673667
    36683668            sage: Partition([1]).outline()
    3669             abs(x - 1) + abs(x + 1) - abs(x)
     3669            abs(x + 1) + abs(x - 1) - abs(x)
    36703670
    36713671            sage: y=sage.symbolic.ring.var("y")
    36723672            sage: Partition([6,5,1]).outline(variable=y)
    3673             abs(y - 3) - abs(y - 2) + abs(y - 1) - abs(y + 3) + abs(y + 4) - abs(y + 5) + abs(y + 6)
     3673            abs(y + 6) - abs(y + 5) + abs(y + 4) - abs(y + 3) + abs(y - 1) - abs(y - 2) + abs(y - 3)
    36743674
    36753675        TESTS::
    36763676
  • sage/combinat/perfect_matching.py

    diff --git a/sage/combinat/perfect_matching.py b/sage/combinat/perfect_matching.py
    a b  
    728728            sage: m = PerfectMatching([(1,3),(2,4)])
    729729            sage: n = PerfectMatching([(1,2),(3,4)])
    730730            sage: factor(m.Weingarten_function(N,n))
    731             -1/((N - 1)*(N + 2)*N)
     731            -1/((N + 2)*(N - 1)*N)
    732732        """
    733733        if other is None:
    734734            other = self.parent().an_element()
  • sage/combinat/q_analogues.py

    diff --git a/sage/combinat/q_analogues.py b/sage/combinat/q_analogues.py
    a b  
    218218
    219219        sage: z = var('z')
    220220        sage: factor(q_binomial(4,2,z))
    221         (z^2 + 1)*(z^2 + z + 1)
     221        (z^2 + z + 1)*(z^2 + 1)
    222222
    223223    This also works for complex roots of unity::
    224224
  • sage/combinat/root_system/plot.py

    diff --git a/sage/combinat/root_system/plot.py b/sage/combinat/root_system/plot.py
    a b  
    12051205    Four vectors in dimension 3::
    12061206
    12071207        sage: m = barycentric_projection_matrix(3); m
    1208         [ 1/3*sqrt(2)*sqrt(3) -1/3*sqrt(2)*sqrt(3)             0   0]
    1209         [         1/3*sqrt(2)          1/3*sqrt(2)  -2/3*sqrt(2)   0]
    1210         [                 1/3                  1/3           1/3  -1]
     1208        [ 1/3*sqrt(3)*sqrt(2) -1/3*sqrt(3)*sqrt(2)                    0                    0]
     1209        [         1/3*sqrt(2)          1/3*sqrt(2)         -2/3*sqrt(2)                    0]
     1210        [                 1/3                  1/3                  1/3                   -1]
    12111211
    12121212    The columns give four vectors that sum up to zero::
    12131213
  • sage/combinat/sf/ns_macdonald.py

    diff --git a/sage/combinat/sf/ns_macdonald.py b/sage/combinat/sf/ns_macdonald.py
    a b  
    534534            sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
    535535            sage: q,t = var('q,t')
    536536            sage: a.coeff(q,t)
    537             (t - 1)^4/((q*t^2 - 1)^2*(q^2*t^3 - 1)^2)
     537            (t - 1)^4/((q^2*t^3 - 1)^2*(q*t^2 - 1)^2)
    538538        """
    539539        res = 1
    540540        shape = self.shape()
     
    554554            sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
    555555            sage: q,t = var('q,t')
    556556            sage: a.coeff_integral(q,t)
    557             (t - 1)^4*(q*t^2 - 1)^2*(q^2*t^3 - 1)^2
     557            (q^2*t^3 - 1)^2*(q*t^2 - 1)^2*(t - 1)^4
    558558        """
    559559        res = 1
    560560        shape = self.shape()
  • sage/combinat/sf/sfa.py

    diff --git a/sage/combinat/sf/sfa.py b/sage/combinat/sf/sfa.py
    a b  
    24472447            490/1539
    24482448            sage: (x,y) = var('x,y')
    24492449            sage: a.scalar_qt(a,q=x,t=y)
    2450             2/3*(x - 1)^3/(y - 1)^3 + 1/3*(x^3 - 1)/(y^3 - 1)
     2450            1/3*(x^3 - 1)/(y^3 - 1) + 2/3*(x - 1)^3/(y - 1)^3
    24512451            sage: Rn = QQ['q','t','y','z'].fraction_field()
    24522452            sage: (q,t,y,z) = Rn.gens()
    24532453            sage: Mac = SymmetricFunctions(Rn).macdonald(q=y,t=z)
  • sage/combinat/tutorial.py

    diff --git a/sage/combinat/tutorial.py b/sage/combinat/tutorial.py
    a b  
    518518    sage: Cf = sage.symbolic.function_factory.function('C')
    519519    sage: equadiff.substitute_function(Cf, s0)
    520520    doctest:...: DeprecationWarning:...
    521     sqrt(-4*z + 1) + (4*z - 1)/sqrt(-4*z + 1) == 0
     521    (4*z - 1)/sqrt(-4*z + 1) + sqrt(-4*z + 1) == 0
    522522    sage: bool(equadiff.substitute_function(Cf, s0))
    523523    True
    524524
     
    11771177    ::
    11781178
    11791179        sage: factor(sum( x^p.length() for p in Permutations(3) ))
    1180         (x + 1)*(x^2 + x + 1)
     1180        (x^2 + x + 1)*(x + 1)
    11811181
    11821182    ::
    11831183
  • sage/ext/fast_callable.pyx

    diff --git a/sage/ext/fast_callable.pyx b/sage/ext/fast_callable.pyx
    a b  
    3030form) at x=30:
    3131
    3232sage: wilk = prod((x-i) for i in [1 .. 20]); wilk
    33 (x - 20)*(x - 19)*(x - 18)*(x - 17)*(x - 16)*(x - 15)*(x - 14)*(x - 13)*(x - 12)*(x - 11)*(x - 10)*(x - 9)*(x - 8)*(x - 7)*(x - 6)*(x - 5)*(x - 4)*(x - 3)*(x - 2)*(x - 1)
     33(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17)*(x - 18)*(x - 19)*(x - 20)
    3434sage: timeit('wilk.subs(x=30)') # random, long time
    3535625 loops, best of 3: 1.43 ms per loop
    3636sage: fc_wilk = fast_callable(wilk, vars=[x])
     
    370370        sage: fp(e, pi, sqrt(2))
    371371        -98.0015640336
    372372        sage: symbolic_result = p(e, pi, sqrt(2)); symbolic_result
    373         -pi^2*e - pi^2 - 3*sqrt(2)*e - 2*e - 6*e^2
     373        -pi^2*e - pi^2 - 3*sqrt(2)*e - 6*e^2 - 2*e
    374374        sage: n(symbolic_result)
    375375        -98.0015640336293
    376376
  • sage/functions/exp_integral.py

    diff --git a/sage/functions/exp_integral.py b/sage/functions/exp_integral.py
    a b  
    609609        EXAMPLES::
    610610
    611611            sage: log_integral_offset(3)
    612             -log_integral(2) + log_integral(3)
     612            log_integral(3) - log_integral(2)
    613613
    614614        """
    615615        BuiltinFunction.__init__(self, "log_integral_offset", nargs=1,
     
    737737        x*sin_integral(x) + cos(x)
    738738
    739739        sage: integrate(sin(x)/x, x)
    740         1/2*I*Ei(-I*x) - 1/2*I*Ei(I*x)
     740        -1/2*I*Ei(I*x) + 1/2*I*Ei(-I*x)
    741741
    742742
    743743    Compare values of the functions `\operatorname{Si}(x)` and
  • sage/functions/hyperbolic.py

    diff --git a/sage/functions/hyperbolic.py b/sage/functions/hyperbolic.py
    a b  
    9595        :meth:`sage.symbolic.expression.Expression.simplify`::
    9696
    9797            sage: sinh(arccosh(x),hold=True).simplify()
    98             sqrt(x - 1)*sqrt(x + 1)
     98            sqrt(x + 1)*sqrt(x - 1)
    9999
    100100        """
    101101        GinacFunction.__init__(self, "sinh", latex_name=r"\sinh")
     
    274274            sage: bool(diff(sech(x), x) == diff(1/cosh(x), x))
    275275            True
    276276            sage: diff(sech(x), x)
    277             -tanh(x)*sech(x)
     277            -sech(x)*tanh(x)
    278278        """
    279279        x = args[0]
    280280        return -sech(x)*tanh(x)
     
    668668        EXAMPLES::
    669669       
    670670            sage: diff(acsch(x), x)
    671             -1/(sqrt(1/x^2 + 1)*x^2)
     671            -1/(x^2*sqrt(1/x^2 + 1))
    672672        """
    673673        x = args[0]
    674674        return -1/(x**2 * (1 + x**(-2)).sqrt())
  • sage/functions/orthogonal_polys.py

    diff --git a/sage/functions/orthogonal_polys.py b/sage/functions/orthogonal_polys.py
    a b  
    433433        sage: gen_legendre_P(2, 0, t) == legendre_P(2, t)
    434434        True
    435435        sage: gen_legendre_P(3, 1, t)
    436         -3/2*sqrt(-t^2 + 1)*(5*t^2 - 1)
     436        -3/2*(5*t^2 - 1)*sqrt(-t^2 + 1)
    437437        sage: gen_legendre_P(4, 3, t)
    438         105*sqrt(-t^2 + 1)*(t^2 - 1)*t
     438        105*(t^2 - 1)*sqrt(-t^2 + 1)*t
    439439        sage: gen_legendre_P(7, 3, I).expand()
    440440        -16695*sqrt(2)
    441441        sage: gen_legendre_P(4, 1, 2.5)
     
    469469        sage: gen_legendre_Q(0, 1, x)
    470470        -1/sqrt(-x^2 + 1)
    471471        sage: gen_legendre_Q(2, 4, x).factor()
    472         48*x/((x - 1)^2*(x + 1)^2)
     472        48*x/((x + 1)^2*(x - 1)^2)
    473473    """
    474474    from sage.functions.all import sqrt
    475475    if m <= n:
  • sage/functions/other.py

    diff --git a/sage/functions/other.py b/sage/functions/other.py
    a b  
    927927            sage: gamma_inc(2,0)
    928928            1
    929929            sage: gamma_inc(1/2,2)
    930             -(erf(sqrt(2)) - 1)*sqrt(pi)
     930            -sqrt(pi)*(erf(sqrt(2)) - 1)
    931931            sage: gamma_inc(1/2,1)
    932             -(erf(1) - 1)*sqrt(pi)
     932            -sqrt(pi)*(erf(1) - 1)
    933933            sage: gamma_inc(1/2,0)
    934934            sqrt(pi)
    935935            sage: gamma_inc(x,0)
     
    15161516            sage: beta(-1/2,-1/2)
    15171517            0
    15181518            sage: beta(x/2,3)
    1519             beta(1/2*x, 3)
     1519            beta(3, 1/2*x)
    15201520            sage: beta(.5,.5)
    15211521            3.14159265358979
    15221522            sage: beta(1,2.0+I)
    15231523            0.400000000000000 - 0.200000000000000*I
    15241524            sage: beta(3,x+I)
    1525             beta(x + I, 3)
     1525            beta(3, x + I)
    15261526
    15271527        Note that the order of arguments does not matter::
    15281528
    15291529            sage: beta(1/2,3*x)
    1530             beta(3*x, 1/2)
     1530            beta(1/2, 3*x)
    15311531
    15321532        The result is symbolic if exact input is given::
    15331533
    15341534            sage: beta(2,1+5*I)
    1535             beta(5*I + 1, 2)
     1535            beta(2, 5*I + 1)
    15361536            sage: beta(2, 2.)
    15371537            0.166666666666667
    15381538            sage: beta(I, 2.)
     
    20182018            sage: var('a')
    20192019            a
    20202020            sage: conjugate(a*sqrt(-2)*sqrt(-3))
    2021             conjugate(a)*conjugate(sqrt(-3))*conjugate(sqrt(-2))
     2021            conjugate(sqrt(-2))*conjugate(sqrt(-3))*conjugate(a)
    20222022
    20232023        Test pickling::
    20242024
  • sage/functions/piecewise.py

    diff --git a/sage/functions/piecewise.py b/sage/functions/piecewise.py
    a b  
    11591159            sage: f(x) = x^2
    11601160            sage: f = Piecewise([[(-1,1),f]])
    11611161            sage: f._fourier_series_helper(3, 1, lambda n: 1)
    1162             -4*cos(pi*x)/pi^2 + cos(2*pi*x)/pi^2 + 1/3
     1162            cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3
    11631163        """
    11641164        from sage.all import pi, sin, cos, srange
    11651165        x = self.default_variable()
     
    11861186            sage: f(x) = x^2
    11871187            sage: f = Piecewise([[(-1,1),f]])
    11881188            sage: f.fourier_series_partial_sum(3,1)
    1189             -4*cos(pi*x)/pi^2 + cos(2*pi*x)/pi^2 + 1/3
     1189            cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3
    11901190            sage: f1(x) = -1
    11911191            sage: f2(x) = 2
    11921192            sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
    11931193            sage: f.fourier_series_partial_sum(3,pi)
    1194             -3*sin(2*x)/pi + 3*sin(x)/pi - 3*cos(x)/pi - 1/4
     1194            -3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4
    11951195        """
    11961196        return self._fourier_series_helper(N, L, lambda n: 1)
    11971197 
     
    12121212            sage: f(x) = x^2
    12131213            sage: f = Piecewise([[(-1,1),f]])
    12141214            sage: f.fourier_series_partial_sum_cesaro(3,1)
    1215             -8/3*cos(pi*x)/pi^2 + 1/3*cos(2*pi*x)/pi^2 + 1/3
     1215            1/3*cos(2*pi*x)/pi^2 - 8/3*cos(pi*x)/pi^2 + 1/3
    12161216            sage: f1(x) = -1
    12171217            sage: f2(x) = 2
    12181218            sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
    12191219            sage: f.fourier_series_partial_sum_cesaro(3,pi)
    1220             -sin(2*x)/pi + 2*sin(x)/pi - 2*cos(x)/pi - 1/4
     1220            -2*cos(x)/pi - sin(2*x)/pi + 2*sin(x)/pi - 1/4
    12211221        """
    12221222        return self._fourier_series_helper(N, L, lambda n: 1-n/N)
    12231223
     
    12381238            sage: f(x) = x^2
    12391239            sage: f = Piecewise([[(-1,1),f]])
    12401240            sage: f.fourier_series_partial_sum_hann(3,1)
    1241             -3*cos(pi*x)/pi^2 + 1/4*cos(2*pi*x)/pi^2 + 1/3
     1241            1/4*cos(2*pi*x)/pi^2 - 3*cos(pi*x)/pi^2 + 1/3
    12421242            sage: f1(x) = -1
    12431243            sage: f2(x) = 2
    12441244            sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
    12451245            sage: f.fourier_series_partial_sum_hann(3,pi)
    1246             -3/4*sin(2*x)/pi + 9/4*sin(x)/pi - 9/4*cos(x)/pi - 1/4
     1246            -9/4*cos(x)/pi - 3/4*sin(2*x)/pi + 9/4*sin(x)/pi - 1/4
    12471247        """
    12481248        from sage.all import cos, pi
    12491249        return self._fourier_series_helper(N, L, lambda n: (1+cos(pi*n/N))/2)
     
    12651265            sage: f(x) = x^2
    12661266            sage: f = Piecewise([[(-1,1),f]])
    12671267            sage: f.fourier_series_partial_sum_filtered(3,1,[1,1,1])
    1268             -4*cos(pi*x)/pi^2 + cos(2*pi*x)/pi^2 + 1/3
     1268            cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3
    12691269            sage: f1(x) = -1
    12701270            sage: f2(x) = 2
    12711271            sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
    12721272            sage: f.fourier_series_partial_sum_filtered(3,pi,[1,1,1])
    1273             -3*sin(2*x)/pi + 3*sin(x)/pi - 3*cos(x)/pi - 1/4
     1273            -3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4
    12741274        """
    12751275        return self._fourier_series_helper(N, L, lambda n: F[n])
    12761276       
     
    15771577            sage: x, s, w = var('x, s, w')
    15781578            sage: f = Piecewise([[(0,1),1],[(1,2), 1-x]])
    15791579            sage: f.laplace(x, s)
    1580             (s + 1)*e^(-2*s)/s^2 - e^(-s)/s + 1/s - e^(-s)/s^2
     1580            -e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2
    15811581            sage: f.laplace(x, w)
    1582             (w + 1)*e^(-2*w)/w^2 - e^(-w)/w + 1/w - e^(-w)/w^2
     1582            -e^(-w)/w + (w + 1)*e^(-2*w)/w^2 + 1/w - e^(-w)/w^2
    15831583           
    15841584        ::
    15851585       
  • sage/functions/special.py

    diff --git a/sage/functions/special.py b/sage/functions/special.py
    a b  
    13551355   
    13561356        sage: x,y = var('x,y')
    13571357        sage: spherical_harmonic(3,2,x,y)
    1358         15/4*sqrt(7/30)*e^(2*I*y)*sin(x)^2*cos(x)/sqrt(pi)
     1358        15/4*sqrt(7/30)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
    13591359        sage: spherical_harmonic(3,2,1,2)
    1360         15/4*sqrt(7/30)*e^(4*I)*sin(1)^2*cos(1)/sqrt(pi)
     1360        15/4*sqrt(7/30)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
    13611361    """
    13621362    _init()
    13631363    return meval("spherical_harmonic(%s,%s,%s,%s)"%(ZZ(m),ZZ(n),x,y))
     
    14751475        sage: z = var("z")
    14761476        sage: # this is still wrong: must be abs(sin(z)) + 2*round(z/pi)
    14771477        sage: elliptic_e(z, 1)
    1478         sin(z) + 2*round(z/pi)
     1478        2*round(z/pi) + sin(z)
    14791479        sage: elliptic_e(z, 0)
    14801480        z
    14811481        sage: elliptic_e(0.5, 0.1)
  • sage/functions/trig.py

    diff --git a/sage/functions/trig.py b/sage/functions/trig.py
    a b  
    204204            sage: bool(diff(sec(x), x) == diff(1/cos(x), x))
    205205            True
    206206            sage: diff(sec(x), x)
    207             tan(x)*sec(x)
     207            sec(x)*tan(x)
    208208        """
    209209        return sec(x)*tan(x)
    210210       
     
    296296            sage: bool(diff(csc(x), x) == diff(1/sin(x), x))
    297297            True
    298298            sage: diff(csc(x), x)
    299             -csc(x)*cot(x)
     299            -cot(x)*csc(x)
    300300        """
    301301        return -csc(x)*cot(x)
    302302   
     
    687687        EXAMPLES::
    688688
    689689            sage: diff(acsc(x), x)
    690             -1/(sqrt(-1/x^2 + 1)*x^2)
     690            -1/(x^2*sqrt(-1/x^2 + 1))
    691691        """
    692692        return -1/(x**2 * (1 - x**(-2)).sqrt())
    693693
     
    750750        EXAMPLES::
    751751       
    752752            sage: diff(asec(x), x)
    753             1/(sqrt(-1/x^2 + 1)*x^2)
     753            1/(x^2*sqrt(-1/x^2 + 1))
    754754        """
    755755        return 1/(x**2 * (1 - x**(-2)).sqrt())
    756756   
  • sage/functions/wigner.py

    diff --git a/sage/functions/wigner.py b/sage/functions/wigner.py
    a b  
    243243        sage: clebsch_gordan(1.5,0.5,1, 1.5,-0.5,1)
    244244        1/2*sqrt(3)
    245245        sage: clebsch_gordan(3/2,1/2,1, -1/2,1/2,0)
    246         -sqrt(1/6)*sqrt(3)
     246        -sqrt(3)*sqrt(1/6)
    247247
    248248    NOTES:
    249249
  • sage/graphs/generic_graph.py

    diff --git a/sage/graphs/generic_graph.py b/sage/graphs/generic_graph.py
    a b  
    13821382            [-1 -1  3 -1]
    13831383            [-1  0 -1  2]
    13841384            sage: M = G.laplacian_matrix(normalized=True); M
    1385             [                   1 -1/6*sqrt(2)*sqrt(3) -1/6*sqrt(2)*sqrt(3)         -1/3*sqrt(3)]
    1386             [-1/6*sqrt(2)*sqrt(3)                    1                 -1/2                    0]
    1387             [-1/6*sqrt(2)*sqrt(3)                 -1/2                    1                    0]
     1385            [                   1 -1/6*sqrt(3)*sqrt(2) -1/6*sqrt(3)*sqrt(2)         -1/3*sqrt(3)]
     1386            [-1/6*sqrt(3)*sqrt(2)                    1                 -1/2                    0]
     1387            [-1/6*sqrt(3)*sqrt(2)                 -1/2                    1                    0]
    13881388            [        -1/3*sqrt(3)                    0                    0                    1]
     1389
    13891390            sage: Graph({0:[],1:[2]}).laplacian_matrix(normalized=True)
    13901391            [ 0  0  0]
    13911392            [ 0  1 -1]
  • sage/gsl/dft.py

    diff --git a/sage/gsl/dft.py b/sage/gsl/dft.py
    a b  
    391391            sage: s = IndexedSequence(A,J)
    392392            sage: s.dct()
    393393            <BLANKLINE>
    394             Indexed sequence: [1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1]
     394            Indexed sequence: [1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1]
    395395            indexed by [0, 1, 2, 3, 4]
    396396        """
    397397        from sage.symbolic.constants import pi
  • sage/interfaces/maxima_abstract.py

    diff --git a/sage/interfaces/maxima_abstract.py b/sage/interfaces/maxima_abstract.py
    a b  
    11931193        We illustrate an automatic coercion::
    11941194       
    11951195            sage: c = b + sqrt(3); c
    1196             sqrt(2) + sqrt(3) + 2.5
     1196            sqrt(3) + sqrt(2) + 2.5
    11971197            sage: type(c)
    11981198            <type 'sage.symbolic.expression.Expression'>
    11991199            sage: d = sqrt(3) + b; d
    1200             sqrt(2) + sqrt(3) + 2.5
     1200            sqrt(3) + sqrt(2) + 2.5
    12011201            sage: type(d)
    12021202            <type 'sage.symbolic.expression.Expression'>
    12031203       
  • sage/interfaces/maxima_lib.py

    diff --git a/sage/interfaces/maxima_lib.py b/sage/interfaces/maxima_lib.py
    a b  
    656656            Is  a  positive or negative?
    657657            sage: assume(a>0)
    658658            sage: integrate(1/(x^3 *(a+b*x)^(1/3)),x)
    659             2/9*sqrt(3)*b^2*arctan(1/3*(2*(b*x + a)^(1/3) + a^(1/3))*sqrt(3)/a^(1/3))/a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)
     659            2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)
    660660            sage: var('x, n')
    661661            (x, n)
    662662            sage: integral(x^n,x)
     
    689689        ::
    690690
    691691            sage: integrate(1 / (1 + abs(x-5)), x, -5, 6)
    692             log(2) + log(11)
     692            log(11) + log(2)
    693693
    694694        ::
    695695
    696696            sage: integrate(1/(1 + abs(x)), x)
    697             1/2*(log(-x + 1) + log(x + 1))*sgn(x) - 1/2*log(-x + 1) + 1/2*log(x + 1)
     697            1/2*(log(x + 1) + log(-x + 1))*sgn(x) + 1/2*log(x + 1) - 1/2*log(-x + 1)
    698698
    699699        ::
    700700
    701701            sage: integrate(cos(x + abs(x)), x)
    702             1/4*(sgn(x) + 1)*sin(2*x) - 1/2*x*sgn(x) + 1/2*x
     702            -1/2*x*sgn(x) + 1/4*(sgn(x) + 1)*sin(2*x) + 1/2*x
    703703
    704704        An example from sage-support thread e641001f8b8d1129::
    705705
     
    715715        ::
    716716
    717717            sage: integrate(sqrt(x + sqrt(x)), x).simplify_full()
    718             1/12*sqrt(sqrt(x) + 1)*((8*x - 3)*x^(1/4) + 2*x^(3/4)) - 1/8*log(sqrt(sqrt(x) + 1) - x^(1/4)) + 1/8*log(sqrt(sqrt(x) + 1) + x^(1/4))
     718            1/12*((8*x - 3)*x^(1/4) + 2*x^(3/4))*sqrt(sqrt(x) + 1) + 1/8*log(sqrt(sqrt(x) + 1) + x^(1/4)) - 1/8*log(sqrt(sqrt(x) + 1) - x^(1/4))
    719719
    720720        And :trac:`11594`::
    721721
  • sage/matrix/matrix2.pyx

    diff --git a/sage/matrix/matrix2.pyx b/sage/matrix/matrix2.pyx
    a b  
    304304            (3, 5)
    305305            sage: soln=A.solve_right(result)
    306306            sage: soln
    307             (-(3*c/a - 5)*b/((b*c/a - d)*a) + 3/a, (3*c/a - 5)/(b*c/a - d))
     307            (-b*(3*c/a - 5)/(a*(b*c/a - d)) + 3/a, (3*c/a - 5)/(b*c/a - d))
    308308            sage: (a*x+b*y).subs(x=soln[0],y=soln[1]).simplify_full()
    309309            3
    310310            sage: (c*x+d*y).subs(x=soln[0],y=soln[1]).simplify_full()
     
    47574757            sage: S = matrix([[x, y], [y, 3*x^2]])
    47584758            sage: em = S.eigenmatrix_left()
    47594759            sage: eigenvalues = em[0]; eigenvalues
    4760             [-1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) + 3/2*x^2 + 1/2*x                                                        0]
    4761             [                                                       0 3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)]
     4760            [3/2*x^2 + 1/2*x - 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)                                                       0]
     4761            [                                                      0 3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)]
    47624762            sage: eigenvectors = em[1]; eigenvectors
    4763             [                                                     1  1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y]
    4764             [                                                     1 -1/2*(x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) - 3*x^2)/y]
     4763            [                                                    1 1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y]
     4764            [                                                    1 1/2*(3*x^2 - x + sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y]
     4765
    47654766
    47664767        A request for ``'all'`` the eigenvalues, when it is not
    47674768        possible, will raise an error.  Using the ``'galois'``
     
    50255026            sage: S = matrix([[x, y], [y, 3*x^2]])
    50265027            sage: em = S.eigenmatrix_right()
    50275028            sage: eigenvalues = em[0]; eigenvalues
    5028             [-1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) + 3/2*x^2 + 1/2*x                                                        0]
    5029             [                                                       0  3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)]
     5029            [3/2*x^2 + 1/2*x - 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)                                                       0]
     5030            [                                                      0 3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)]
     5031
    50305032            sage: eigenvectors = em[1]; eigenvectors
    5031             [                                                     1                                                      1]
    5032             [ 1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y -1/2*(x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) - 3*x^2)/y]
     5033            [                                                    1                                                     1]
     5034            [1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y 1/2*(3*x^2 - x + sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y]
     5035
    50335036
    50345037        TESTS::
    50355038
     
    1157511578       
    1157611579            sage: a=matrix([[1,2],[3,4]])
    1157711580            sage: a.exp()
    11578             [-1/22*((sqrt(33) - 11)*e^sqrt(33) - sqrt(33) - 11)*e^(-1/2*sqrt(33) + 5/2)              2/33*(sqrt(33)*e^sqrt(33) - sqrt(33))*e^(-1/2*sqrt(33) + 5/2)]
    11579             [             1/11*(sqrt(33)*e^sqrt(33) - sqrt(33))*e^(-1/2*sqrt(33) + 5/2)  1/22*((sqrt(33) + 11)*e^sqrt(33) - sqrt(33) + 11)*e^(-1/2*sqrt(33) + 5/2)]
     11581            [-1/22*(e^sqrt(33)*(sqrt(33) - 11) - sqrt(33) - 11)*e^(-1/2*sqrt(33) + 5/2)              2/33*(sqrt(33)*e^sqrt(33) - sqrt(33))*e^(-1/2*sqrt(33) + 5/2)]
     11582            [             1/11*(sqrt(33)*e^sqrt(33) - sqrt(33))*e^(-1/2*sqrt(33) + 5/2)  1/22*(e^sqrt(33)*(sqrt(33) + 11) - sqrt(33) + 11)*e^(-1/2*sqrt(33) + 5/2)]
     11583
    1158011584            sage: type(a.exp())
    1158111585            <type 'sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense'>
    1158211586
    1158311587            sage: a=matrix([[1/2,2/3],[3/4,4/5]])
    1158411588            sage: a.exp()
    11585             [-1/418*((3*sqrt(209) - 209)*e^(1/10*sqrt(209)) - 3*sqrt(209) - 209)*e^(-1/20*sqrt(209) + 13/20)                   20/627*(sqrt(209)*e^(1/10*sqrt(209)) - sqrt(209))*e^(-1/20*sqrt(209) + 13/20)]
    11586             [                  15/418*(sqrt(209)*e^(1/10*sqrt(209)) - sqrt(209))*e^(-1/20*sqrt(209) + 13/20)  1/418*((3*sqrt(209) + 209)*e^(1/10*sqrt(209)) - 3*sqrt(209) + 209)*e^(-1/20*sqrt(209) + 13/20)]
     11589            [-1/418*(e^(1/10*sqrt(209))*(3*sqrt(209) - 209) - 3*sqrt(209) - 209)*e^(-1/20*sqrt(209) + 13/20)                   20/627*(sqrt(209)*e^(1/10*sqrt(209)) - sqrt(209))*e^(-1/20*sqrt(209) + 13/20)]
     11590            [                  15/418*(sqrt(209)*e^(1/10*sqrt(209)) - sqrt(209))*e^(-1/20*sqrt(209) + 13/20)  1/418*(e^(1/10*sqrt(209))*(3*sqrt(209) + 209) - 3*sqrt(209) + 209)*e^(-1/20*sqrt(209) + 13/20)]
    1158711591
    1158811592            sage: a=matrix(RR,[[1,pi.n()],[1e2,1e-2]])
    1158911593            sage: a.exp()
    11590             [ 1/11882424341266*((11*sqrt(227345670387496707609) + 5941212170633)*e^(3/1275529100*sqrt(227345670387496707609)) - 11*sqrt(227345670387496707609) + 5941212170633)*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200)                            445243650/75781890129165569203*(sqrt(227345670387496707609)*e^(3/1275529100*sqrt(227345670387496707609)) - sqrt(227345670387496707609))*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200)]
    11591             [                                     10000/53470909535697*(sqrt(227345670387496707609)*e^(3/1275529100*sqrt(227345670387496707609)) - sqrt(227345670387496707609))*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200) -1/11882424341266*((11*sqrt(227345670387496707609) - 5941212170633)*e^(3/1275529100*sqrt(227345670387496707609)) - 11*sqrt(227345670387496707609) - 5941212170633)*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200)]
     11594            [ 1/11882424341266*(e^(3/1275529100*sqrt(227345670387496707609))*(11*sqrt(227345670387496707609) + 5941212170633) - 11*sqrt(227345670387496707609) + 5941212170633)*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200)                            445243650/75781890129165569203*(sqrt(227345670387496707609)*e^(3/1275529100*sqrt(227345670387496707609)) - sqrt(227345670387496707609))*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200)]
     11595            [                                     10000/53470909535697*(sqrt(227345670387496707609)*e^(3/1275529100*sqrt(227345670387496707609)) - sqrt(227345670387496707609))*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200) -1/11882424341266*(e^(3/1275529100*sqrt(227345670387496707609))*(11*sqrt(227345670387496707609) - 5941212170633) - 11*sqrt(227345670387496707609) - 5941212170633)*e^(-3/2551058200*sqrt(227345670387496707609) + 101/200)]
    1159211596            sage: a.change_ring(RDF).exp()
    1159311597            [42748127.3153 7368259.24416]
    1159411598            [234538976.138 40426191.4516]
  • sage/matrix/matrix_symbolic_dense.pyx

    diff --git a/sage/matrix/matrix_symbolic_dense.pyx b/sage/matrix/matrix_symbolic_dense.pyx
    a b  
    4040
    4141    sage: M = matrix(SR, 2, var('a,b,c,d'))
    4242    sage: ~M
    43     [-b*c/((b*c/a - d)*a^2) + 1/a            b/((b*c/a - d)*a)]
    44     [           c/((b*c/a - d)*a)               -1/(b*c/a - d)]
     43    [1/a - b*c/(a^2*(b*c/a - d))           b/(a*(b*c/a - d))]
     44    [          c/(a*(b*c/a - d))              -1/(b*c/a - d)]
    4545    sage: (~M*M).simplify_rational()
    4646    [1 0]
    4747    [0 1]
     
    103103    sage: t = var('t')
    104104    sage: M = matrix(SR, 2, 2, [cos(t), sin(t), -sin(t), cos(t)])
    105105    sage: M.det()
    106     sin(t)^2 + cos(t)^2
     106    cos(t)^2 + sin(t)^2
    107107
    108108Permanents::
    109109
     
    206206            [3 4 5]
    207207            [6 7 8]
    208208            sage: es = A.eigenvectors_left(); es
    209             [(-3*sqrt(6) + 6, [(1, -1/5*sqrt(6) + 4/5, -2/5*sqrt(2)*sqrt(3) + 3/5)], 1), (3*sqrt(6) + 6, [(1, 1/5*sqrt(6) + 4/5, 2/5*sqrt(2)*sqrt(3) + 3/5)], 1), (0, [(1, -2, 1)], 1)]
     209            [(-3*sqrt(6) + 6, [(1, -1/5*sqrt(6) + 4/5, -2/5*sqrt(3)*sqrt(2) + 3/5)], 1), (3*sqrt(6) + 6, [(1, 1/5*sqrt(6) + 4/5, 2/5*sqrt(3)*sqrt(2) + 3/5)], 1), (0, [(1, -2, 1)], 1)]
    210210            sage: eval, [evec], mult = es[0]
    211211            sage: delta = eval*evec - evec*A
    212212            sage: abs(abs(delta)) < 1e-10
    213             abs(sqrt(144/25*(sqrt(2)*sqrt(3) - sqrt(6))^2 + 1/25*(3*(sqrt(6) - 2)*(2*sqrt(2)*sqrt(3) - 3) + 16*sqrt(2)*sqrt(3) + 5*sqrt(6) - 54)^2 + 1/25*(3*(sqrt(6) - 4)*(sqrt(6) - 2) + 14*sqrt(2)*sqrt(3) + 4*sqrt(6) - 42)^2)) < (1.00000000000000e-10)
     213            abs(sqrt(1/25*(3*(2*sqrt(3)*sqrt(2) - 3)*(sqrt(6) - 2) + 16*sqrt(3)*sqrt(2) + 5*sqrt(6) - 54)^2 + 1/25*(3*(sqrt(6) - 2)*(sqrt(6) - 4) + 14*sqrt(3)*sqrt(2) + 4*sqrt(6) - 42)^2 + 144/25*(sqrt(3)*sqrt(2) - sqrt(6))^2)) < (1.00000000000000e-10)
    214214            sage: abs(abs(delta)).n() < 1e-10
    215215            True
    216216
     
    218218
    219219            sage: A = matrix(SR, 2, 2, var('a,b,c,d'))
    220220            sage: A.eigenvectors_left()
    221             [(1/2*a + 1/2*d - 1/2*sqrt(a^2 - 2*a*d + 4*b*c + d^2), [(1, -1/2*(a - d + sqrt(a^2 - 2*a*d + 4*b*c + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 - 2*a*d + 4*b*c + d^2), [(1, -1/2*(a - d - sqrt(a^2 - 2*a*d + 4*b*c + d^2))/c)], 1)]
     221            [(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
    222222            sage: es = A.eigenvectors_left(); es
    223             [(1/2*a + 1/2*d - 1/2*sqrt(a^2 - 2*a*d + 4*b*c + d^2), [(1, -1/2*(a - d + sqrt(a^2 - 2*a*d + 4*b*c + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 - 2*a*d + 4*b*c + d^2), [(1, -1/2*(a - d - sqrt(a^2 - 2*a*d + 4*b*c + d^2))/c)], 1)]
     223            [(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
    224224            sage: eval, [evec], mult = es[0]
    225225            sage: delta = eval*evec - evec*A
    226226            sage: delta.apply_map(lambda x: x.full_simplify())
     
    400400
    401401            sage: M = matrix(SR, 2, 2, var('a,b,c,d'))
    402402            sage: M.charpoly('t')
    403             t^2 + (-a - d)*t + a*d - b*c
     403            t^2 + (-a - d)*t - b*c + a*d
    404404            sage: matrix(SR, 5, [1..5^2]).charpoly()
    405405            x^5 - 65*x^4 - 250*x^3
    406406
     
    525525            sage: theta = var('theta')
    526526            sage: M = matrix(SR, 2, 2, [cos(theta), sin(theta), -sin(theta), cos(theta)])
    527527            sage: ~M
    528             [-sin(theta)^2/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta)^2) + 1/cos(theta)                    -sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))]
    529             [                    sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))                                           1/(sin(theta)^2/cos(theta) + cos(theta))]
     528            [1/cos(theta) - sin(theta)^2/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta)^2)                   -sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))]
     529            [                   sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))                                          1/(sin(theta)^2/cos(theta) + cos(theta))]
    530530            sage: (~M).simplify_trig()
    531531            [ cos(theta) -sin(theta)]
    532532            [ sin(theta)  cos(theta)]
     
    539539       
    540540            sage: M = matrix(SR, 3, 3, range(9)) - var('t')
    541541            sage: (~M*M)[0,0]
    542             -(3*((6/t + 7)/((t - 3/t - 4)*t) + 2/t)*((6/t + 5)/((t - 3/t - 4)*t) + 2/t)/((6/t + 5)*(6/t + 7)/(t - 3/t - 4) - t + 12/t + 8) - 1/t - 3/((t - 3/t - 4)*t^2))*t + 3*(6/t + 7)*((6/t + 5)/((t - 3/t - 4)*t) + 2/t)/((t - 3/t - 4)*((6/t + 5)*(6/t + 7)/(t - 3/t - 4) - t + 12/t + 8)) + 6*((6/t + 5)/((t - 3/t - 4)*t) + 2/t)/((6/t + 5)*(6/t + 7)/(t - 3/t - 4) - t + 12/t + 8) - 3/((t - 3/t - 4)*t)
     542            t*(3*(2/t + (6/t + 7)/((t - 3/t - 4)*t))*(2/t + (6/t + 5)/((t - 3/t
     543            - 4)*t))/(t - (6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) + 1/t +
     544            3/((t - 3/t - 4)*t^2)) - 6*(2/t + (6/t + 5)/((t - 3/t - 4)*t))/(t -
     545            (6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) - 3*(6/t + 7)*(2/t +
     546            (6/t + 5)/((t - 3/t - 4)*t))/((t - (6/t + 7)*(6/t + 5)/(t - 3/t -
     547            4) - 12/t - 8)*(t - 3/t - 4)) - 3/((t - 3/t - 4)*t)
    543548            sage: expand((~M*M)[0,0])
    544549            1
    545550            sage: (~M * M).simplify_rational()
     
    646651            sage: M = MatrixSpace(SR,2,2)
    647652            sage: h = M(sin(x)+cos(x))
    648653            sage: h
    649             [sin(x) + cos(x)               0]
    650             [              0 sin(x) + cos(x)]
     654            [cos(x) + sin(x)               0]
     655            [              0 cos(x) + sin(x)]
    651656            sage: h(x=1)
    652             [sin(1) + cos(1)               0]
    653             [              0 sin(1) + cos(1)]
     657            [cos(1) + sin(1)               0]
     658            [              0 cos(1) + sin(1)]
    654659            sage: h(x=x)
    655             [sin(x) + cos(x)               0]
    656             [              0 sin(x) + cos(x)]
     660            [cos(x) + sin(x)               0]
     661            [              0 cos(x) + sin(x)]
    657662
    658663            sage: h = M((sin(x)+cos(x)).function(x))
    659664            sage: h
    660             [sin(x) + cos(x)               0]
    661             [              0 sin(x) + cos(x)]
     665            [cos(x) + sin(x)               0]
     666            [              0 cos(x) + sin(x)]
    662667            sage: h(1)
    663668            doctest:...: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)
    664669            See http://trac.sagemath.org/4513 for details.
    665670            doctest:...: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)
    666671            See http://trac.sagemath.org/5930 for details.
    667             [sin(1) + cos(1)               0]
    668             [              0 sin(1) + cos(1)]
     672            [cos(1) + sin(1)               0]
     673            [              0 cos(1) + sin(1)]
    669674            sage: h(x)
    670             [sin(x) + cos(x)               0]
    671             [              0 sin(x) + cos(x)]
     675            [cos(x) + sin(x)               0]
     676            [              0 cos(x) + sin(x)]
    672677
    673678            sage: f = M([0,x,y,z]); f
    674679            [0 x]
  • sage/misc/functional.py

    diff --git a/sage/misc/functional.py b/sage/misc/functional.py
    a b  
    578578    ::
    579579
    580580        sage: sum(k * binomial(n, k), k, 1, n)
    581         n*2^(n - 1)
     581        2^(n - 1)*n
    582582
    583583    ::
    584584
     
    707707        sage: integral(sin(x)^2, x, algorithm='maxima')
    708708        1/2*x - 1/4*sin(2*x)
    709709        sage: integral(sin(x)^2, x, algorithm='sympy')
    710         -1/2*sin(x)*cos(x) + 1/2*x
     710        -1/2*cos(x)*sin(x) + 1/2*x
    711711
    712712    TESTS:
    713713
  • sage/misc/parser.pyx

    diff --git a/sage/misc/parser.pyx b/sage/misc/parser.pyx
    a b  
    451451
    452452            sage: p = Parser(make_var=var)
    453453            sage: p.parse("a*b^c - 3a")
    454             b^c*a - 3*a
     454            a*b^c - 3*a
    455455           
    456456            sage: R.<x> = QQ[]
    457457            sage: p = Parser(make_var = {'x': x })
     
    752752            sage: p.p_term(Tokenizer("-a * b + c"))
    753753            -a*b
    754754            sage: p.p_term(Tokenizer("a*(b-c)^2"))
    755             (b - c)^2*a
     755            a*(b - c)^2
    756756            sage: p.p_term(Tokenizer("-3a"))
    757757            -3*a
    758758        """
  • sage/modules/free_module_element.pyx

    diff --git a/sage/modules/free_module_element.pyx b/sage/modules/free_module_element.pyx
    a b  
    17751775       
    17761776            sage: x = var('x')
    17771777            sage: v = vector([x/(2*x)+sqrt(2)+var('theta')^3,x/(2*x)]); v
    1778             (sqrt(2) + theta^3 + 1/2, 1/2)
     1778            (theta^3 + sqrt(2) + 1/2, 1/2)
    17791779            sage: v._repr_()
    1780             '(sqrt(2) + theta^3 + 1/2, 1/2)'
     1780            '(theta^3 + sqrt(2) + 1/2, 1/2)'
    17811781        """
    17821782        d = self.degree()
    17831783        if d == 0: return "()"
     
    33313331            [   (r, theta) |--> cos(theta) (r, theta) |--> -r*sin(theta)]
    33323332            [   (r, theta) |--> sin(theta)  (r, theta) |--> r*cos(theta)]
    33333333            sage: T.diff().det() # Jacobian
    3334             (r, theta) |--> r*sin(theta)^2 + r*cos(theta)^2
     3334            (r, theta) |--> r*cos(theta)^2 + r*sin(theta)^2
    33353335        """
    33363336        if var is None:
    33373337            if sage.symbolic.callable.is_CallableSymbolicExpressionRing(self.base_ring()):
  • sage/modules/vector_callable_symbolic_dense.py

    diff --git a/sage/modules/vector_callable_symbolic_dense.py b/sage/modules/vector_callable_symbolic_dense.py
    a b  
    1818    sage: 3*f
    1919    (r, theta, z) |--> (3*r*cos(theta), 3*r*sin(theta), 3*z)
    2020    sage: f*f # dot product
    21     (r, theta, z) |--> r^2*sin(theta)^2 + r^2*cos(theta)^2 + z^2
     21    (r, theta, z) |--> r^2*cos(theta)^2 + r^2*sin(theta)^2 + z^2
    2222    sage: f.diff()(0,1,2) # the matrix derivative
    2323    [cos(1)      0      0]
    2424    [sin(1)      0      0]
  • sage/modules/vector_symbolic_dense.py

    diff --git a/sage/modules/vector_symbolic_dense.py b/sage/modules/vector_symbolic_dense.py
    a b  
    1313
    1414    sage: x, y = var('x, y')
    1515    sage: u = vector([sin(x)^2 + cos(x)^2, log(2*y) + log(3*y)]); u
    16     (sin(x)^2 + cos(x)^2, log(2*y) + log(3*y))
     16    (cos(x)^2 + sin(x)^2, log(3*y) + log(2*y))
    1717    sage: type(u)
    1818    <class 'sage.modules.vector_symbolic_dense.Vector_symbolic_dense'>
    1919    sage: u.simplify_full()
     
    8484            sage: v.simplify_trig()
    8585            (1, log(x*y), sin(1/(x + 1)), factorial(x + 1)/factorial(x))
    8686            sage: v.simplify_radical()
    87             (sin(x)^2 + cos(x)^2, log(x) + log(y), sin(1/(x + 1)), factorial(x + 1)/factorial(x))
     87            (cos(x)^2 + sin(x)^2, log(x) + log(y), sin(1/(x + 1)), factorial(x + 1)/factorial(x))
    8888            sage: v.simplify_rational()
    89             (sin(x)^2 + cos(x)^2, log(x*y), sin(1/(x + 1)), factorial(x + 1)/factorial(x))
     89            (cos(x)^2 + sin(x)^2, log(x*y), sin(1/(x + 1)), factorial(x + 1)/factorial(x))
    9090            sage: v.simplify_factorial()
    91             (sin(x)^2 + cos(x)^2, log(x*y), sin(x/(x^2 + x)), x + 1)
     91            (cos(x)^2 + sin(x)^2, log(x*y), sin(x/(x^2 + x)), x + 1)
    9292            sage: v.simplify_full()
    9393            (1, log(x*y), sin(1/(x + 1)), x + 1)
    9494
    9595            sage: v = vector([sin(2*x), sin(3*x)])
    9696            sage: v.simplify_trig()
    97             (2*sin(x)*cos(x), (4*cos(x)^2 - 1)*sin(x))
     97            (2*cos(x)*sin(x), (4*cos(x)^2 - 1)*sin(x))
    9898            sage: v.simplify_trig(False)
    9999            (sin(2*x), sin(3*x))
    100100            sage: v.simplify_trig(expand=False)
  • sage/plot/plot3d/plot3d.py

    diff --git a/sage/plot/plot3d/plot3d.py b/sage/plot/plot3d/plot3d.py
    a b  
    416416
    417417        sage: r, phi, theta = var('r phi theta')
    418418        sage: T.transform(radius=r, azimuth=theta, inclination=phi)
    419         (r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi))
     419        (r*cos(theta)*sin(phi), r*sin(phi)*sin(theta), r*cos(phi))
    420420   
    421421    We can plot with this transform.  Remember that the dependent
    422422    variable is the radius, and the independent variables are the
     
    442442       
    443443            sage: T = Spherical('radius', ['azimuth', 'inclination'])
    444444            sage: T.transform(radius=var('r'), azimuth=var('theta'), inclination=var('phi'))
    445             (r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi))
     445            (r*cos(theta)*sin(phi), r*sin(phi)*sin(theta), r*cos(phi))
    446446        """
    447447        return (radius * sin(inclination) * cos(azimuth),
    448448                radius * sin(inclination) * sin(azimuth),
     
    471471        sage: T = SphericalElevation('radius', ['azimuth', 'elevation'])
    472472        sage: r, theta, phi = var('r theta phi')
    473473        sage: T.transform(radius=r, azimuth=theta, elevation=phi)
    474         (r*cos(phi)*cos(theta), r*sin(theta)*cos(phi), r*sin(phi))
     474        (r*cos(phi)*cos(theta), r*cos(phi)*sin(theta), r*sin(phi))
    475475
    476476    We can plot with this transform.  Remember that the dependent
    477477    variable is the radius, and the independent variables are the
     
    519519
    520520            sage: T = SphericalElevation('radius', ['azimuth', 'elevation'])
    521521            sage: T.transform(radius=var('r'), azimuth=var('theta'), elevation=var('phi'))
    522             (r*cos(phi)*cos(theta), r*sin(theta)*cos(phi), r*sin(phi))
     522            (r*cos(phi)*cos(theta), r*cos(phi)*sin(theta), r*sin(phi))
    523523        """
    524524        return (radius * cos(elevation) * cos(azimuth),
    525525                radius * cos(elevation) * sin(azimuth),
  • sage/rings/integer.pyx

    diff --git a/sage/rings/integer.pyx b/sage/rings/integer.pyx
    a b  
    19281928            sage: 2^I                # complex number
    19291929            2^I
    19301930            sage: f = 2^(sin(x)-cos(x)); f
    1931             2^(sin(x) - cos(x))
     1931            2^(-cos(x) + sin(x))
    19321932            sage: f(x=3)
    1933             2^(sin(3) - cos(3))
     1933            2^(-cos(3) + sin(3))
    19341934       
    19351935        A symbolic sum::
    19361936       
  • sage/rings/number_field/number_field_element.pyx

    diff --git a/sage/rings/number_field/number_field_element.pyx b/sage/rings/number_field/number_field_element.pyx
    a b  
    20712071       
    20722072            sage: K.<a> = NumberField(x^3 + x - 1, embedding=0.68)
    20732073            sage: b = SR(a); b # indirect doctest
    2074             1/3*(3*(1/18*sqrt(3)*sqrt(31) + 1/2)^(2/3) - 1)/(1/18*sqrt(3)*sqrt(31) + 1/2)^(1/3)
     2074            1/3*(3*(1/18*sqrt(31)*sqrt(3) + 1/2)^(2/3) - 1)/(1/18*sqrt(31)*sqrt(3) + 1/2)^(1/3)
    20752075
    20762076            sage: (b^3 + b - 1).simplify_radical()
    20772077            0
  • sage/rings/polynomial/polynomial_element.pyx

    diff --git a/sage/rings/polynomial/polynomial_element.pyx b/sage/rings/polynomial/polynomial_element.pyx
    a b  
    46684668       
    46694669            sage: X = var('X')
    46704670            sage: f = expand((X-1)*(X-I)^3*(X^2 - sqrt(2))); f
    4671             -sqrt(2)*X^4 + I*sqrt(2) + X^6 - (3*I + 1)*X^5 + (3*I - 3)*X^4 + (3*I + 1)*sqrt(2)*X^3 + (I + 3)*X^3 - (3*I - 3)*sqrt(2)*X^2 - I*X^2 - (I + 3)*sqrt(2)*X
     4671            X^6 - (3*I + 1)*X^5 - sqrt(2)*X^4 + (3*I - 3)*X^4 + (3*I + 1)*sqrt(2)*X^3 + (I + 3)*X^3 - (3*I - 3)*sqrt(2)*X^2 - I*X^2 - (I + 3)*sqrt(2)*X + I*sqrt(2)
    46724672            sage: print f.roots()
    46734673            [(I, 3), (-2^(1/4), 1), (2^(1/4), 1), (1, 1)]
    46744674       
  • sage/rings/power_series_ring.py

    diff --git a/sage/rings/power_series_ring.py b/sage/rings/power_series_ring.py
    a b  
    5555    sage: a, b, c = var('a,b,c')
    5656    sage: f = a + b*t + c*t^2 + O(t^3)
    5757    sage: f*f
    58     a^2 + 2*a*b*t + (2*a*c + b^2)*t^2 + O(t^3)
     58    a^2 + 2*a*b*t + (b^2 + 2*a*c)*t^2 + O(t^3)
    5959    sage: f = sqrt(2) + sqrt(3)*t + O(t^3)
    6060    sage: f^2
    61     2 + 2*sqrt(2)*sqrt(3)*t + 3*t^2 + O(t^3)
     61    2 + 2*sqrt(3)*sqrt(2)*t + 3*t^2 + O(t^3)
    6262
    6363Elements are first coerced to constants in base_ring, then coerced
    6464into the PowerSeriesRing::
  • sage/rings/qqbar.py

    diff --git a/sage/rings/qqbar.py b/sage/rings/qqbar.py
    a b  
    619619
    620620            sage: x = polygen(SR)
    621621            sage: p = (x - sqrt(-5)) * (x - sqrt(3)); p
    622             x^2 + (-sqrt(-5) - sqrt(3))*x + sqrt(-5)*sqrt(3)
     622            x^2 + (-sqrt(3) - sqrt(-5))*x + sqrt(3)*sqrt(-5)
    623623            sage: p = QQbar.common_polynomial(p)
    624624            sage: a = QQbar.polynomial_root(p, CIF(RIF(-0.1, 0.1), RIF(2, 3))); a
    625625            0.?e-18 + 2.236067977499790?*I
  • sage/schemes/elliptic_curves/ell_generic.py

    diff --git a/sage/schemes/elliptic_curves/ell_generic.py b/sage/schemes/elliptic_curves/ell_generic.py
    a b  
    424424            0
    425425
    426426            sage: 2*w
    427             (-2*pi + (2*pi - 3*pi^2 + 10)^2/(-40*pi + 4*pi^3 - 4*pi^2 - 79) + 1 : (2*pi - 3*pi^2 + 10)*(3*pi - (2*pi - 3*pi^2 + 10)^2/(-40*pi + 4*pi^3 - 4*pi^2 - 79) - 1)/sqrt(-40*pi + 4*pi^3 - 4*pi^2 - 79) + 1/2*sqrt(-40*pi + 4*pi^3 - 4*pi^2 - 79) - 1/2 : 1)
     427            (-2*pi + (2*pi - 3*pi^2 + 10)^2/(-40*pi + 4*pi^3 - 4*pi^2 - 79) + 1 : (3*pi - (2*pi - 3*pi^2 + 10)^2/(-40*pi + 4*pi^3 - 4*pi^2 - 79) - 1)*(2*pi - 3*pi^2 + 10)/sqrt(-40*pi + 4*pi^3 - 4*pi^2 - 79) + 1/2*sqrt(-40*pi + 4*pi^3 - 4*pi^2 - 79) - 1/2 : 1)
    428428           
    429429            sage: x, y, z = 2*w; temp = ((y^2 + y) - (x^3 - x^2 - 10*x - 20))
    430430
  • sage/stats/basic_stats.py

    diff --git a/sage/stats/basic_stats.py b/sage/stats/basic_stats.py
    a b  
    165165        sage: std([])
    166166        NaN
    167167        sage: std([I, sqrt(2), 3/5])
    168         sqrt(1/450*(5*sqrt(2) + 5*I - 6)^2 + 1/450*(5*sqrt(2) - 10*I + 3)^2 + 1/450*(10*sqrt(2) - 5*I - 3)^2)
     168        sqrt(1/450*(10*sqrt(2) - 5*I - 3)^2 + 1/450*(5*sqrt(2) - 10*I + 3)^2 + 1/450*(5*sqrt(2) + 5*I - 6)^2)
    169169        sage: std([RIF(1.0103, 1.0103), RIF(2)])
    170170        0.6998235813403261?
    171171        sage: import numpy
     
    230230        sage: variance([])
    231231        NaN
    232232        sage: variance([I, sqrt(2), 3/5])
    233         1/450*(5*sqrt(2) + 5*I - 6)^2 + 1/450*(5*sqrt(2) - 10*I + 3)^2 + 1/450*(10*sqrt(2) - 5*I - 3)^2
     233        1/450*(10*sqrt(2) - 5*I - 3)^2 + 1/450*(5*sqrt(2) - 10*I + 3)^2 + 1/450*(5*sqrt(2) + 5*I - 6)^2
    234234        sage: variance([RIF(1.0103, 1.0103), RIF(2)])
    235235        0.4897530450000000?
    236236        sage: import numpy
  • sage/symbolic/callable.py

    diff --git a/sage/symbolic/callable.py b/sage/symbolic/callable.py
    a b  
    1515    sage: f(x, y, z) = sin(x+y+z)
    1616    sage: g(w, t) = cos(w - t)
    1717    sage: f + g
    18     (t, w, x, y, z) |--> sin(x + y + z) + cos(-t + w)
     18    (t, w, x, y, z) |--> cos(-t + w) + sin(x + y + z)
    1919
    2020::
    2121
  • sage/symbolic/constants.py

    diff --git a/sage/symbolic/constants.py b/sage/symbolic/constants.py
    a b  
    130130    sage: f^2
    131131    (I*e + I)^2
    132132    sage: _.expand()
    133     -2*e - e^2 - 1
     133    -e^2 - 2*e - 1
    134134   
    135135::
    136136
     
    180180floating point rings::
    181181
    182182    sage: a = pi + e + golden_ratio + log2 + euler_gamma + catalan + khinchin + twinprime + mertens; a
    183     pi + euler_gamma + catalan + golden_ratio + log2 + khinchin + twinprime + mertens + e
     183    mertens + twinprime + khinchin + log2 + golden_ratio + catalan + euler_gamma + pi + e
    184184    sage: parent(a)
    185185    Symbolic Ring
    186     sage: RR(a)
     186    sage: RR(a) #abstol 1e11
    187187    13.2713479401972
    188188    sage: RealField(212)(a)
    189189    13.2713479401972493100988191995758139408711068200030748178329712
    190190    sage: RealField(230)(a)
    191191    13.271347940197249310098819199575813940871106820003074817832971189555
    192     sage: CC(a)
     192    sage: CC(a) #abstol 1e11
    193193    13.2713479401972
    194194    sage: CDF(a)
    195195    13.2713479402
  • sage/symbolic/expression.pyx

    diff --git a/sage/symbolic/expression.pyx b/sage/symbolic/expression.pyx
    a b  
    7878    sage: var('a,b,c')
    7979    (a, b, c)
    8080    sage: expand((u + v + a + b + c)^2)
    81     a^2 + 2*a*b + 2*a*c + 2*a*u + 2*a*v + b^2 + 2*b*c + 2*b*u + 2*b*v + c^2 + 2*c*u + 2*c*v + u^2 + 2*u*v + v^2
     81    a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2 + 2*a*u + 2*b*u + 2*c*u + u^2 + 2*a*v + 2*b*v + 2*c*v + 2*u*v + v^2
    8282
    8383TESTS:
    8484
     
    108108    sage: t*u
    109109    1
    110110    sage: t + u
    111     e^sqrt(x) + e^(-sqrt(x))
     111    e^(-sqrt(x)) + e^sqrt(x)
    112112    sage: t
    113113    e^sqrt(x)
    114114
     
    120120    sage: a.real_part()
    121121    4*sqrt(3)/(sqrt(3) + 5)
    122122    sage: a.imag_part()
    123     sqrt(abs(4*(sqrt(3) - 5)*(sqrt(3) + 5) + 48))/(sqrt(3) + 5)
     123    sqrt(abs(4*(sqrt(3) + 5)*(sqrt(3) - 5) + 48))/(sqrt(3) + 5)
    124124"""
    125125
    126126###############################################################################
     
    327327            sage: t = 2*x*y^z+3
    328328            sage: u = loads(dumps(t)) # indirect doctest
    329329            sage: u
    330             2*y^z*x + 3
     330            2*x*y^z + 3
    331331            sage: bool(t == u)
    332332            True
    333333            sage: u.subs(x=z)
     
    529529        EXAMPLES::
    530530       
    531531            sage: gap(e + pi^2 + x^3)
    532             pi^2 + x^3 + e
     532            x^3 + pi^2 + e
    533533        """
    534534        return '"%s"'%repr(self)
    535535
     
    541541        EXAMPLES::
    542542       
    543543            sage: singular(e + pi^2 + x^3)
    544             pi^2 + x^3 + e
     544            x^3 + pi^2 + e
    545545        """
    546546        return '"%s"'%repr(self)
    547547
     
    557557            sage: x = var('x')                     
    558558            sage: f = sin(cos(x^2) + log(x))
    559559            sage: f._magma_init_(magma)
    560             '"sin(log(x) + cos(x^2))"'
     560            '"sin(cos(x^2) + log(x))"'
    561561            sage: magma(f)                         # optional - magma
    562562            sin(log(x) + cos(x^2))
    563563            sage: magma(f).Type()                  # optional - magma
     
    576576            sage: latex(y + 3*(x^(-1)))
    577577            y + \frac{3}{x}
    578578            sage: latex(x^(y+z^(1/y)))
    579             x^{z^{\left(\frac{1}{y}\right)} + y}
     579            x^{y + z^{\left(\frac{1}{y}\right)}}
    580580            sage: latex(1/sqrt(x+y))
    581581            \frac{1}{\sqrt{x + y}}
    582582            sage: latex(sin(x*(z+y)^x))
    583             \sin\left({\left(y + z\right)}^{x} x\right)
     583            \sin\left(x {\left(y + z\right)}^{x}\right)
    584584            sage: latex(3/2*(x+y)/z/y)
    585585            \frac{3 \, {\left(x + y\right)}}{2 \, y z}
    586586            sage: latex((2^(x^y)))
     
    590590            sage: latex((x*y).conjugate())
    591591            \overline{x} \overline{y}
    592592            sage: latex(x*(1/(x^2)+sqrt(x^7)))
    593             {\left(\sqrt{x^{7}} + \frac{1}{x^{2}}\right)} x
     593            x {\left(\sqrt{x^{7}} + \frac{1}{x^{2}}\right)}
    594594
    595595        Check spacing of coefficients of mul expressions (#3202)::
    596596
     
    653653            sage: latex((x+2)/(x^3+1))
    654654            \frac{x + 2}{x^{3} + 1}
    655655            sage: latex((x+2)*(x+1)/(x^3+1))
    656             \frac{{\left(x + 1\right)} {\left(x + 2\right)}}{x^{3} + 1}
     656            \frac{{\left(x + 2\right)} {\left(x + 1\right)}}{x^{3} + 1}
    657657            sage: latex((x+2)/(x^3+1)/(x+1))
    658             \frac{x + 2}{{\left(x + 1\right)} {\left(x^{3} + 1\right)}}
     658            \frac{x + 2}{{\left(x^{3} + 1\right)} {\left(x + 1\right)}}
    659659
    660660        Check that the sign is correct (#9086)::
    661661
     
    12361236            sage: x^3 -y == y + x
    12371237            x^3 - y == x + y
    12381238            sage: x^3 - y^10 >= y + x^10
    1239             x^3 - y^10 >= x^10 + y
     1239            -y^10 + x^3 >= x^10 + y
    12401240            sage: x^2 > x
    12411241            x^2 > x
    12421242
     
    13911391            sage: v,c = var('v,c')
    13921392            sage: assume(c != 0)
    13931393            sage: integral((1+v^2/c^2)^3/(1-v^2/c^2)^(3/2),v)
    1394             -17/8*v^3/(sqrt(-v^2/c^2 + 1)*c^2) - 1/4*v^5/(sqrt(-v^2/c^2 + 1)*c^4) + 83/8*v/sqrt(-v^2/c^2 + 1) - 75/8*arcsin(v/(c^2*sqrt(c^(-2))))/sqrt(c^(-2))
     1394            83/8*v/sqrt(-v^2/c^2 + 1) - 17/8*v^3/(c^2*sqrt(-v^2/c^2 + 1)) - 1/4*v^5/(c^4*sqrt(-v^2/c^2 + 1)) - 75/8*arcsin(v/(c^2*sqrt(c^(-2))))/sqrt(c^(-2))
    13951395            sage: forget()
    13961396        """
    13971397        from sage.symbolic.assumptions import _assumptions
     
    25082508
    25092509            # check if comparison of constant terms in Pynac add objects work
    25102510            sage: (y-1)*(y-2)
    2511             (y - 2)*(y - 1)
     2511            (y - 1)*(y - 2)
    25122512
    25132513        Check if Pynac can compute inverses of Python longs (:trac:`13107`)::
    25142514
     
    27572757       
    27582758            sage: x,y = var('x,y')
    27592759            sage: x.__cmp__(y)
    2760             -1
     2760            1
    27612761            sage: x < y
    27622762            x < y
    27632763            sage: cmp(x,y)
    2764             -1
     2764            1
    27652765            sage: cmp(SR(0.5), SR(0.7))
    27662766            -1
    27672767            sage: SR(0.5) < SR(0.7)
     
    31103110       
    31113111            sage: h = sin(x)/cos(x)
    31123112            sage: derivative(h,x,x,x)
    3113             6*sin(x)^4/cos(x)^4 + 8*sin(x)^2/cos(x)^2 + 2
     3113            8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
    31143114            sage: derivative(h,x,3)
    3115             6*sin(x)^4/cos(x)^4 + 8*sin(x)^2/cos(x)^2 + 2
     3115            8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
    31163116
    31173117        ::
    31183118       
     
    31203120            (x, y)
    31213121            sage: u = (sin(x) + cos(y))*(cos(x) - sin(y))
    31223122            sage: derivative(u,x,y)
    3123             sin(x)*sin(y) - cos(x)*cos(y)           
     3123            -cos(x)*cos(y) + sin(x)*sin(y)
    31243124            sage: f = ((x^2+1)/(x^2-1))^(1/4)
    31253125            sage: g = derivative(f, x); g # this is a complex expression
    3126             1/2*(x/(x^2 - 1) - (x^2 + 1)*x/(x^2 - 1)^2)/((x^2 + 1)/(x^2 - 1))^(3/4)
     3126            -1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4)
    31273127            sage: g.factor()
    3128             -x/((x - 1)^2*(x + 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))
     3128            -x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))
    31293129 
    31303130        ::
    31313131       
    31323132            sage: y = var('y')
    31333133            sage: f = y^(sin(x))
    31343134            sage: derivative(f, x)
    3135             y^sin(x)*log(y)*cos(x)
     3135            y^sin(x)*cos(x)*log(y)
    31363136       
    31373137        ::
    31383138       
     
    31503150       
    31513151            sage: g = 1/(sqrt((x^2-1)*(x+5)^6))
    31523152            sage: derivative(g, x)
    3153             -((x + 5)^6*x + 3*(x + 5)^5*(x^2 - 1))/((x + 5)^6*(x^2 - 1))^(3/2)
     3153            -((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)
    31543154
    31553155        TESTS::
    31563156       
     
    33243324            sage: g = f.series(x==1, 4); g
    33253325            (-sin(y) - 1) + (-2*sin(y) - 2)*(x - 1) + (-sin(y) + 3)*(x - 1)^2 + 1*(x - 1)^3
    33263326            sage: h = g.truncate(); h
    3327             -(sin(y) - 3)*(x - 1)^2 + (x - 1)^3 - 2*(sin(y) + 1)*(x - 1) - sin(y) - 1
     3327            (x - 1)^3 - (x - 1)^2*(sin(y) - 3) - 2*(x - 1)*(sin(y) + 1) - sin(y) - 1
    33283328            sage: h.expand()
    33293329            x^3 - x^2*sin(y) - 5*x + 3
    33303330
     
    33343334            sage: f.series(x,7)
    33353335            1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7)
    33363336            sage: f.series(x==1,3)
    3337             (sin(1)) + (-2*sin(1) + cos(1))*(x - 1) + (5/2*sin(1) - 2*cos(1))*(x - 1)^2 + Order((x - 1)^3)
     3337            (sin(1)) + (cos(1) - 2*sin(1))*(x - 1) + (-2*cos(1) + 5/2*sin(1))*(x - 1)^2 + Order((x - 1)^3)
    33383338            sage: f.series(x==1,3).truncate().expand()
    3339             5/2*x^2*sin(1) - 2*x^2*cos(1) - 7*x*sin(1) + 5*x*cos(1) + 11/2*sin(1) - 3*cos(1)
     3339            -2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
    33403340
    33413341        Following the GiNaC tutorial, we use John Machin's amazing
    33423342        formula `\pi = 16 \tan^{-1}(1/5) - 4 \tan^{-1}(1/239)` to compute
     
    33873387            sage: var('a, x, z')
    33883388            (a, x, z)
    33893389            sage: taylor(a*log(z), z, 2, 3)
    3390             1/24*(z - 2)^3*a - 1/8*(z - 2)^2*a + 1/2*(z - 2)*a + a*log(2)
     3390            1/24*a*(z - 2)^3 - 1/8*a*(z - 2)^2 + 1/2*a*(z - 2) + a*log(2)
    33913391
    33923392        ::
    33933393
     
    34243424        Check that ticket #7472 is fixed (Taylor polynomial in more variables)::
    34253425 
    34263426            sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,1),4)
    3427             (y - 1)^3*(x - 1) + (y - 1)^3 + 3*(y - 1)^2*(x - 1) + 3*(y - 1)^2 + 3*(y - 1)*(x - 1) + x + 3*y - 3
     3427            (x - 1)*(y - 1)^3 + 3*(x - 1)*(y - 1)^2 + (y - 1)^3 + 3*(x - 1)*(y - 1) + 3*(y - 1)^2 + x + 3*y - 3
    34283428            sage: expand(_)
    34293429            x*y^3
    34303430
     
    34693469            sage: f.series(x,7).truncate()
    34703470            -1/5040*x^5 + 1/120*x^3 - 1/6*x + 1/x
    34713471            sage: f.series(x==1,3).truncate().expand()
    3472             5/2*x^2*sin(1) - 2*x^2*cos(1) - 7*x*sin(1) + 5*x*cos(1) + 11/2*sin(1) - 3*cos(1)
     3472            -2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
    34733473        """
    34743474        if not is_a_series(self._gobj):
    34753475            return self
     
    35203520            sage: ((x + (2/3)*y)^3).expand()
    35213521            x^3 + 2*x^2*y + 4/3*x*y^2 + 8/27*y^3
    35223522            sage: expand( (x*sin(x) - cos(y)/x)^2 )
    3523             x^2*sin(x)^2 - 2*sin(x)*cos(y) + cos(y)^2/x^2
     3523            x^2*sin(x)^2 - 2*cos(y)*sin(x) + cos(y)^2/x^2
    35243524            sage: f = (x-y)*(x+y); f
    3525             (x - y)*(x + y)
     3525            (x + y)*(x - y)
    35263526            sage: f.expand()
    35273527            x^2 - y^2
    35283528        """
     
    35793579        EXAMPLES::
    35803580       
    35813581            sage: sin(5*x).expand_trig()
    3582             sin(x)^5 - 10*sin(x)^3*cos(x)^2 + 5*sin(x)*cos(x)^4
     3582            5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5
    35833583            sage: cos(2*x + var('y')).expand_trig()
    3584             -sin(2*x)*sin(y) + cos(2*x)*cos(y)
     3584            cos(2*x)*cos(y) - sin(2*x)*sin(y)
    35853585       
    35863586        We illustrate various options to this function::
    35873587       
    35883588            sage: f = sin(sin(3*cos(2*x))*x)
    35893589            sage: f.expand_trig()
    3590             sin(-(sin(cos(2*x))^3 - 3*sin(cos(2*x))*cos(cos(2*x))^2)*x)
     3590            sin((3*cos(cos(2*x))^2*sin(cos(2*x)) - sin(cos(2*x))^3)*x)
    35913591            sage: f.expand_trig(full=True)
    3592             sin(((sin(sin(x)^2)*cos(cos(x)^2) - sin(cos(x)^2)*cos(sin(x)^2))^3 - 3*(sin(sin(x)^2)*cos(cos(x)^2) - sin(cos(x)^2)*cos(sin(x)^2))*(sin(sin(x)^2)*sin(cos(x)^2) + cos(sin(x)^2)*cos(cos(x)^2))^2)*x)
     3592            sin((3*(cos(cos(x)^2)*cos(sin(x)^2) + sin(cos(x)^2)*sin(sin(x)^2))^2*(cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2)) - (cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2))^3)*x)
    35933593            sage: sin(2*x).expand_trig(times=False)
    35943594            sin(2*x)
    35953595            sage: sin(2*x).expand_trig(times=True)
    3596             2*sin(x)*cos(x)
     3596            2*cos(x)*sin(x)
    35973597            sage: sin(2 + x).expand_trig(plus=False)
    35983598            sin(x + 2)
    35993599            sage: sin(2 + x).expand_trig(plus=True)
    3600             sin(2)*cos(x) + sin(x)*cos(2)
     3600            cos(x)*sin(2) + cos(2)*sin(x)
    36013601            sage: sin(x/2).expand_trig(half_angles=False)
    36023602            sin(1/2*x)
    36033603            sage: sin(x/2).expand_trig(half_angles=True)
    3604             sqrt(-1/2*cos(x) + 1/2)*(-1)^floor(1/2*x/pi)
     3604            (-1)^floor(1/2*x/pi)*sqrt(-1/2*cos(x) + 1/2)
    36053605
    36063606        ALIASES:
    36073607
     
    36423642            sage: y=var('y')
    36433643            sage: f=sin(x)*cos(x)^3+sin(y)^2
    36443644            sage: f.reduce_trig()
    3645             1/4*sin(2*x) + 1/8*sin(4*x) - 1/2*cos(2*y) + 1/2
     3645            -1/2*cos(2*y) + 1/8*sin(4*x) + 1/4*sin(2*x) + 1/2
    36463646
    36473647        To reduce only the expressions involving x we use optional parameter::
    36483648
    36493649            sage: f.reduce_trig(x)
    3650             sin(y)^2 + 1/4*sin(2*x) + 1/8*sin(4*x)
     3650            sin(y)^2 + 1/8*sin(4*x) + 1/4*sin(2*x)
    36513651
    36523652        ALIASES: :meth:`trig_reduce` and :meth:`reduce_trig` are the same
    36533653        """
     
    37033703            {$0: x + y}
    37043704            sage: t = ((a+b)*(a+c)).match((a+w0)*(a+w1))
    37053705            sage: t[w0], t[w1]
    3706             (b, c)
     3706            (c, b)
    37073707            sage: ((a+b)*(a+c)).match((w0+b)*(w0+c))
    37083708            {$0: a}
    3709             sage: print ((a+b)*(a+c)).match((w0+w1)*(w0+w2))    # surprising?
     3709            sage: t = ((a+b)*(a+c)).match((w0+w1)*(w0+w2))
     3710            sage: t[w0], t[w1], t[w2]
     3711            (a, c, b)
     3712            sage: print ((a+b)*(a+c)).match((w0+w1)*(w1+w2))
    37103713            None
    37113714            sage: t = (a*(x+y)+a*z+b).match(a*w0+w1)
    37123715            sage: t[w0], t[w1]
     
    37723775            sage: w0 = SR.wild(0); w1 = SR.wild(1)
    37733776
    37743777            sage: (sin(x)*sin(y)).find(sin(w0))
    3775             [sin(x), sin(y)]
     3778            [sin(y), sin(x)]
    37763779
    37773780            sage: ((sin(x)+sin(y))*(a+b)).expand().find(sin(w0))
    3778             [sin(x), sin(y)]
     3781            [sin(y), sin(x)]
    37793782
    37803783            sage: (1+x+x^2+x^3).find(x)
    37813784            [x]
    37823785            sage: (1+x+x^2+x^3).find(x^w0)
    3783             [x^3, x^2]
     3786            [x^2, x^3]
    37843787
    37853788            sage: (1+x+x^2+x^3).find(y)
    37863789            []
     
    37973800        while itr.is_not_equal(found.end()):
    37983801            res.append(new_Expression_from_GEx(self._parent, itr.obj()))
    37993802            itr.inc()
     3803        res.sort(cmp)
    38003804        return res
    38013805
    38023806    def has(self, pattern):
     
    38673871            (x + y)^3 + b^2 + c
    38683872
    38693873            sage: t.subs({w0^2: w0^3})
    3870             (x + y)^3 + a^3 + b^3
     3874            a^3 + b^3 + (x + y)^3
    38713875
    38723876            # substitute with a relational expression
    38733877            sage: t.subs(w0^2 == w0^3)
    3874             (x + y)^3 + a^3 + b^3
     3878            a^3 + b^3 + (x + y)^3
    38753879
    38763880            sage: t.subs(w0==w0^2)
    38773881            (x^2 + y^2)^18 + a^16 + b^16           
     
    39893993            (x, y, z, a, b, c, d, f)
    39903994            sage: w0 = SR.wild(0); w1 = SR.wild(1)
    39913995            sage: (a^2 + b^2 + (x+y)^2)._subs_expr(w0^2 == w0^3)
    3992             (x + y)^3 + a^3 + b^3
     3996            a^3 + b^3 + (x + y)^3
    39933997            sage: (a^4 + b^4 + (x+y)^4)._subs_expr(w0^2 == w0^3)
    3994             (x + y)^4 + a^4 + b^4
     3998            a^4 + b^4 + (x + y)^4
    39953999            sage: (a^2 + b^4 + (x+y)^4)._subs_expr(w0^2 == w0^3)
    3996             (x + y)^4 + a^3 + b^4
     4000            b^4 + (x + y)^4 + a^3
    39974001            sage: ((a+b+c)^2)._subs_expr(a+b == x)
    39984002            (a + b + c)^2
    39994003            sage: ((a+b+c)^2)._subs_expr(a+b+w0 == x+w0)
     
    40174021            sage: (sin(x)^2 + cos(x)^2)._subs_expr(sin(w0)^2+cos(w0)^2==1)
    40184022            1
    40194023            sage: (1 + sin(x)^2 + cos(x)^2)._subs_expr(sin(w0)^2+cos(w0)^2==1)
    4020             sin(x)^2 + cos(x)^2 + 1
     4024            cos(x)^2 + sin(x)^2 + 1
    40214025            sage: (17*x + sin(x)^2 + cos(x)^2)._subs_expr(w1 + sin(w0)^2+cos(w0)^2 == w1 + 1)
    40224026            17*x + 1
    40234027            sage: ((x-1)*(sin(x)^2 + cos(x)^2)^2)._subs_expr(sin(w0)^2+cos(w0)^2 == 1)
     
    40644068            x^4 + x
    40654069            sage: f = cos(x^2) + sin(x^2)
    40664070            sage: f.subs_expr(x^2 == x)
    4067             sin(x) + cos(x)
     4071            cos(x) + sin(x)
    40684072       
    40694073        ::
    40704074       
    40714075            sage: f(x,y,t) = cos(x) + sin(y) + x^2 + y^2 + t
    40724076            sage: f.subs_expr(y^2 == t)
    4073             (x, y, t) |--> x^2 + 2*t + sin(y) + cos(x)
     4077            (x, y, t) |--> x^2 + 2*t + cos(x) + sin(y)
    40744078       
    40754079        The following seems really weird, but it *is* what Maple does::
    40764080       
    40774081            sage: f.subs_expr(x^2 + y^2 == t)
    4078             (x, y, t) |--> x^2 + y^2 + t + sin(y) + cos(x)
     4082            (x, y, t) |--> x^2 + y^2 + t + cos(x) + sin(y)
    40794083            sage: maple.eval('subs(x^2 + y^2 = t, cos(x) + sin(y) + x^2 + y^2 + t)')          # optional - maple
    40804084            'cos(x)+sin(y)+x^2+y^2+t'
    40814085            sage: maxima.quit()
     
    41244128            sage: var('x,y,z')
    41254129            (x, y, z)
    41264130            sage: (x+y)(x=z^2, y=x^y)
    4127             x^y + z^2
     4131            z^2 + x^y
    41284132        """
    41294133        return self._parent._call_element_(self, *args, **kwds)
    41304134
     
    42624266            sage: var('a,b,c,x,y')
    42634267            (a, b, c, x, y)
    42644268            sage: (a^2 + b^2 + (x+y)^2).operands()
    4265             [(x + y)^2, a^2, b^2]
     4269            [a^2, b^2, (x + y)^2]
    42664270            sage: (a^2).operands()
    42674271            [a, 2]
    42684272            sage: (a*b^2*c).operands()
     
    46384642            sage: f
    46394643            (x, y) |--> x^n + y^n
    46404644            sage: f(2,3)
    4641             2^n + 3^n
     4645            3^n + 2^n
    46424646        """
    46434647        # we override type checking in CallableSymbolicExpressionRing,
    46444648        # since it checks for old SymbolicVariable's
     
    46954699            sage: x.add(x, hold=True)
    46964700            x + x
    46974701            sage: x.add(x, (2+x), hold=True)
    4698             x + x + (x + 2)
     4702            (x + 2) + x + x
    46994703            sage: x.add(x, (2+x), x, hold=True)
    4700             x + x + (x + 2) + x
     4704            (x + 2) + x + x + x
    47014705            sage: x.add(x, (2+x), x, 2*x, hold=True)
    4702             x + x + (x + 2) + x + 2*x
     4706            (x + 2) + 2*x + x + x + x
    47034707
    47044708        To then evaluate again, we currently must use Maxima via
    47054709        :meth:`simplify`::
     
    47284732            sage: x.mul(x, hold=True)
    47294733            x*x
    47304734            sage: x.mul(x, (2+x), hold=True)
    4731             x*x*(x + 2)
     4735            (x + 2)*x*x
    47324736            sage: x.mul(x, (2+x), x, hold=True)
    4733             x*x*(x + 2)*x
     4737            (x + 2)*x*x*x
    47344738            sage: x.mul(x, (2+x), x, 2*x, hold=True)
    4735             x*x*(x + 2)*x*(2*x)
     4739            (2*x)*(x + 2)*x*x*x
    47364740
    47374741        To then evaluate again, we currently must use Maxima via
    47384742        :meth:`simplify`::
     
    47924796            sage: f.coefficient(sin(x*y))
    47934797            x^3 + 2/x
    47944798            sage: f.collect(sin(x*y))
    4795             (x^3 + 2/x)*sin(x*y) + a*x + x*y + x/y + 100
     4799            a*x + x*y + (x^3 + 2/x)*sin(x*y) + x/y + 100
    47964800
    47974801            sage: var('a, x, y, z')
    47984802            (a, x, y, z)
     
    49794983            sage: bool(p.poly(a) == (x-a*sqrt(2))^2 + x + 1)
    49804984            True           
    49814985            sage: p.poly(x)
    4982             -(2*sqrt(2)*a - 1)*x + 2*a^2 + x^2 + 1
     4986            2*a^2 - (2*sqrt(2)*a - 1)*x + x^2 + 1
    49834987        """
    49844988        from sage.symbolic.ring import SR
    49854989        f = self._maxima_()
     
    51255129            sage: R = SR[x]
    51265130            sage: a = R(sqrt(2) + x^3 + y)
    51275131            sage: a
    5128             y + sqrt(2) + x^3
     5132            x^3 + y + sqrt(2)
    51295133            sage: type(a)
    51305134            <class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field'>
    51315135            sage: a.degree()
     
    52955299            sage: lcm(x^100-y^100, x^10-y^10)
    52965300            -x^100 + y^100
    52975301            sage: lcm(expand( (x^2+17*x+3/7*y)*(x^5 - 17*y + 2/3) ), expand((x^13+17*x+3/7*y)*(x^5 - 17*y + 2/3)) )
    5298             1/21*(21*x^7 + 357*x^6 + 9*x^5*y - 357*x^2*y + 14*x^2 - 6069*x*y - 153*y^2 + 238*x + 6*y)*(21*x^18 - 357*x^13*y + 14*x^13 + 357*x^6 + 9*x^5*y - 6069*x*y - 153*y^2 + 238*x + 6*y)/(3*x^5 - 51*y + 2)
     5302             1/21*(21*x^18 - 357*x^13*y + 14*x^13 + 357*x^6 + 9*x^5*y -
     5303                     6069*x*y - 153*y^2 + 238*x + 6*y)*(21*x^7 + 357*x^6 +
     5304                             9*x^5*y - 357*x^2*y + 14*x^2 - 6069*x*y -
     5305                             153*y^2 + 238*x + 6*y)/(3*x^5 - 51*y + 2)
    52995306           
    53005307        TESTS:
    53015308       
     
    53505357            sage: x,y,z = var('x,y,z')
    53515358            sage: f = 4*x*y + x*z + 20*y^2 + 21*y*z + 4*z^2 + x^2*y^2*z^2
    53525359            sage: f.collect(x)
    5353             x^2*y^2*z^2 + (4*y + z)*x + 20*y^2 + 21*y*z + 4*z^2
     5360            x^2*y^2*z^2 + x*(4*y + z) + 20*y^2 + 21*y*z + 4*z^2
    53545361
    53555362        Here we do the same thing for `y` and `z`; however, note that
    53565363        we don't factor the `y^{2}` and `z^{2}` terms before
     
    53595366            sage: f.collect(y)
    53605367            (x^2*z^2 + 20)*y^2 + (4*x + 21*z)*y + x*z + 4*z^2
    53615368            sage: f.collect(z)
    5362             (x^2*y^2 + 4)*z^2 + (x + 21*y)*z + 4*x*y + 20*y^2
     5369            (x^2*y^2 + 4)*z^2 + 4*x*y + 20*y^2 + (x + 21*y)*z
    53635370
    53645371        Sometimes, we do have to call :meth:`expand()` on the
    53655372        expression first to achieve the desired result::
     
    53685375            sage: f.collect(x)
    53695376            x^2 + x*y - x*z - y*z
    53705377            sage: f.expand().collect(x)
    5371             (y - z)*x + x^2 - y*z
     5378            x^2 + x*(y - z) - y*z
    53725379
    53735380        TESTS:
    53745381
     
    57495756            (a, b)
    57505757            sage: f = log(a + b*I)
    57515758            sage: f.imag_part()
    5752             arctan2(real_part(b) + imag_part(a), real_part(a) - imag_part(b))
     5759            arctan2(imag_part(a) + real_part(b), -imag_part(b) + real_part(a))
    57535760
    57545761        Using the ``hold`` parameter it is possible to prevent automatic
    57555762        evaluation::
     
    62856292        To prevent automatic evaluation use the ``hold`` argument::
    62866293
    62876294            sage: arccosh(x).sinh()
    6288             sqrt(x - 1)*sqrt(x + 1)
     6295            sqrt(x + 1)*sqrt(x - 1)
    62896296            sage: arccosh(x).sinh(hold=True)
    62906297            sinh(arccosh(x))
    62916298
     
    62946301            sage: sinh(arccosh(x),hold=True)
    62956302            sinh(arccosh(x))
    62966303            sage: sinh(arccosh(x))
    6297             sqrt(x - 1)*sqrt(x + 1)
     6304            sqrt(x + 1)*sqrt(x - 1)
    62986305
    62996306        To then evaluate again, we currently must use Maxima via
    63006307        :meth:`simplify`::
    63016308
    63026309            sage: a = arccosh(x).sinh(hold=True); a.simplify()
    6303             sqrt(x - 1)*sqrt(x + 1)
     6310            sqrt(x + 1)*sqrt(x - 1)
    63046311
    63056312        TESTS::
    63066313
     
    67746781            sage: x.factorial()
    67756782            factorial(x)
    67766783            sage: (x^2+y^3).factorial()
    6777             factorial(x^2 + y^3)
     6784            factorial(y^3 + x^2)
    67786785
    67796786        To prevent automatic evaluation use the ``hold`` argument::
    67806787
     
    70687075
    70697076            sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a
    70707077            sage: f.normalize()
    7071             (a*x^3 + a*x*y^2 + b*x^3 + c*x^3 + a*x^2 + a*y^2 + b*x^2 + c*x^2 - a*x - 7*b*x - 7*c*x - 7*a - 7*b - 7*c)/((x + 1)*(x^2 - 7)*a)
     7078            (a*x^3 + b*x^3 + c*x^3 + a*x*y^2 + a*x^2 + b*x^2 + c*x^2 +
     7079                    a*y^2 - a*x - 7*b*x - 7*c*x - 7*a - 7*b - 7*c)/((x^2 -
     7080                        7)*a*(x + 1))
    70727081
    70737082        ALGORITHM: Uses GiNaC.
    70747083
     
    71907199            sage: f.numerator()
    71917200            sqrt(x) + sqrt(y) + sqrt(z)
    71927201            sage: f.denominator()
    7193             -sqrt(theta) + x^10 - y^10
     7202            x^10 - y^10 - sqrt(theta)
    71947203
    71957204            sage: f.numerator(normalize=False)
    7196             -(sqrt(x) + sqrt(y) + sqrt(z))
     7205            (sqrt(x) + sqrt(y) + sqrt(z))
    71977206            sage: f.denominator(normalize=False)
    7198             sqrt(theta) - x^10 + y^10
     7207            x^10 - y^10 - sqrt(theta)
    71997208
    72007209            sage: y = var('y')
    72017210            sage: g = x + y/(x + 2); g
     
    74037412        EXAMPLES::
    74047413
    74057414            sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
    7406             log(sqrt(2) - 1) + log(sqrt(2) + 1)
     7415            log(sqrt(2) + 1) + log(sqrt(2) - 1)
    74077416            sage: res = t.maxima_methods().logcontract(); res
    7408             log((sqrt(2) - 1)*(sqrt(2) + 1))
     7417            log((sqrt(2) + 1)*(sqrt(2) - 1))
    74097418            sage: type(res)
    74107419            <type 'sage.symbolic.expression.Expression'>
    74117420        """
     
    74657474
    74667475            sage: f = e^(I*x)
    74677476            sage: f.rectform()
    7468             I*sin(x) + cos(x)
     7477            cos(x) + I*sin(x)
    74697478
    74707479        TESTS:
    74717480
     
    75927601        EXAMPLES::
    75937602       
    75947603            sage: f = sin(x)^2 + cos(x)^2; f
    7595             sin(x)^2 + cos(x)^2
     7604            cos(x)^2 + sin(x)^2
    75967605            sage: f.simplify()
    7597             sin(x)^2 + cos(x)^2
     7606            cos(x)^2 + sin(x)^2
    75987607            sage: f.simplify_trig()
    75997608            1
    76007609            sage: h = sin(x)*csc(x)
     
    76657674        ::
    76667675       
    76677676            sage: f = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)); f
    7668             ((x - 1)^(3/2) - sqrt(x - 1)*(x + 1))/sqrt((x - 1)*(x + 1))
     7677            -((x + 1)*sqrt(x - 1) - (x - 1)^(3/2))/sqrt((x + 1)*(x - 1))
    76697678            sage: f.simplify_rational()
    76707679            -2*sqrt(x - 1)/sqrt(x^2 - 1)
    76717680
     
    76897698            sage: y = var('y')
    76907699            sage: g = (x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1)
    76917700            sage: g.simplify_rational(algorithm='simple')
    7692             -(2*x^y - x^(2*y) - 1)/(x^y - 1)
     7701            (x^(2*y) - 2*x^y + 1)/(x^y - 1)
    76937702            sage: g.simplify_rational()
    76947703            x^y - 1
    76957704
     
    77007709            sage: f.simplify_rational()
    77017710            (2*x^2 + 5*x + 4)/(x^3 + 5*x^2 + 8*x + 4)
    77027711            sage: f.simplify_rational(algorithm='noexpand')
    7703             ((x + 1)*x + (x + 2)^2)/((x + 1)*(x + 2)^2)
    7704 
     7712            ((x + 2)^2 + (x + 1)*x)/((x + 2)^2*(x + 1))
    77057713        """
    77067714        self_m = self._maxima_()
    77077715        if algorithm == 'full':
     
    77537761        ::
    77547762
    77557763            sage: f = binomial(n, k)*factorial(k)*factorial(n-k); f
    7756             factorial(-k + n)*factorial(k)*binomial(n, k)
     7764            binomial(n, k)*factorial(k)*factorial(-k + n)
    77577765            sage: f.simplify_factorial()
    77587766            factorial(n)
    77597767       
    77607768        A more complicated example, which needs further processing::
    77617769
    77627770            sage: f = factorial(x)/factorial(x-2)/2 + factorial(x+1)/factorial(x)/2; f
    7763             1/2*factorial(x)/factorial(x - 2) + 1/2*factorial(x + 1)/factorial(x)
     7771            1/2*factorial(x + 1)/factorial(x) + 1/2*factorial(x)/factorial(x - 2)
    77647772            sage: g = f.simplify_factorial(); g
    77657773            1/2*(x - 1)*x + 1/2*x + 1/2
    77667774            sage: g.simplify_rational()
     
    78477855            sage: e1 = 1/(sqrt(5)+sqrt(2))
    78487856            sage: e2 = (sqrt(5)-sqrt(2))/3
    78497857            sage: e1.simplify_radical()
    7850             1/(sqrt(2) + sqrt(5))
     7858            1/(sqrt(5) + sqrt(2))
    78517859            sage: e2.simplify_radical()
    7852             -1/3*sqrt(2) + 1/3*sqrt(5)
     7860            1/3*sqrt(5) - 1/3*sqrt(2)
    78537861            sage: (e1-e2).simplify_radical()
    78547862            0
    78557863        """
     
    79467954
    79477955            sage: f = log(x)+log(y)-1/3*log((x+1))
    79487956            sage: f.simplify_log()
    7949             -1/3*log(x + 1) + log(x*y)
     7957            log(x*y) - 1/3*log(x + 1)
    79507958
    79517959            sage: f.simplify_log('ratios')
    79527960            log(x*y/(x + 1)^(1/3))
     
    79727980
    79737981            sage: log_expr = (log(sqrt(2)-1)+log(sqrt(2)+1))
    79747982            sage: log_expr.simplify_log('all')
    7975             log((sqrt(2) - 1)*(sqrt(2) + 1))
     7983            log((sqrt(2) + 1)*(sqrt(2) - 1))
    79767984            sage: _.simplify_rational()
    79777985            0
    79787986            sage: log_expr.simplify_full()   # applies both simplify_log and simplify_rational
     
    80678075        To expand also log(3/4) use ``algorithm='all'``::
    80688076
    80698077            sage: (log(3/4*x^pi)).log_expand('all')
    8070             pi*log(x) + log(3) - log(4)
     8078            pi*log(x) - log(4) + log(3)
    80718079
    80728080        To expand only the power use ``algorithm='powers'``.::
    80738081
     
    80908098            pi*log(x) + log(3/4)
    80918099
    80928100            sage: (log(3/4*x^pi)).log_expand('all')
    8093             pi*log(x) + log(3) - log(4)
     8101            pi*log(x) - log(4) + log(3)
    80948102
    80958103            sage: (log(3/4*x^pi)).log_expand()
    80968104            pi*log(x) + log(3/4)
     
    81618169       
    81628170            sage: x,y,z = var('x, y, z')
    81638171            sage: (x^3-y^3).factor()
    8164             (x - y)*(x^2 + x*y + y^2)
     8172            (x^2 + x*y + y^2)*(x - y)
    81658173            sage: factor(-8*y - 4*x + z^2*(2*y + x))
    8166             (z - 2)*(z + 2)*(x + 2*y)
     8174            (x + 2*y)*(z + 2)*(z - 2)
    81678175            sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2
    81688176            sage: F = factor(f/(36*(1 + 2*y + y^2)), dontfactor=[x]); F
    8169             1/36*(y - 1)*(x^2 + 2*x + 1)/(y + 1)
     8177            1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1)
    81708178
    81718179        If you are factoring a polynomial with rational coefficients (and
    81728180        dontfactor is empty) the factorization is done using Singular
     
    81768184            sage: var('x,y')
    81778185            (x, y)
    81788186            sage: (x^99 + y^99).factor()
    8179             (x + y)*(x^2 - x*y + y^2)*(x^6 - x^3*y^3 + y^6)*...
     8187            (x^60 + x^57*y^3 - x^51*y^9 - x^48*y^12 + x^42*y^18 + x^39*y^21 -
     8188            x^33*y^27 - x^30*y^30 - x^27*y^33 + x^21*y^39 + x^18*y^42 -
     8189            x^12*y^48 - x^9*y^51 + x^3*y^57 + y^60)*(x^20 + x^19*y -
     8190            x^17*y^3 - x^16*y^4 + x^14*y^6 + x^13*y^7 - x^11*y^9 -
     8191            x^10*y^10 - x^9*y^11 + x^7*y^13 + x^6*y^14 - x^4*y^16 -
     8192            x^3*y^17 + x*y^19 + y^20)*(x^10 - x^9*y + x^8*y^2 - x^7*y^3 +
     8193            x^6*y^4 - x^5*y^5 + x^4*y^6 - x^3*y^7 + x^2*y^8 - x*y^9 +
     8194            y^10)*(x^6 - x^3*y^3 + y^6)*(x^2 - x*y + y^2)*(x + y)
    81808195        """
    81818196        from sage.calculus.calculus import symbolic_expression_from_maxima_string, symbolic_expression_from_string
    81828197        if len(dontfactor) > 0:
     
    82178232            (x, y, z)
    82188233            sage: f = x^3-y^3
    82198234            sage: f.factor()
    8220             (x - y)*(x^2 + x*y + y^2)
     8235            (x^2 + x*y + y^2)*(x - y)
    82218236       
    82228237        Notice that the -1 factor is separated out::
    82238238       
    82248239            sage: f.factor_list()
    8225             [(x - y, 1), (x^2 + x*y + y^2, 1)]
     8240            [(x^2 + x*y + y^2, 1), (x - y, 1)]
    82268241       
    82278242        We factor a fairly straightforward expression::
    82288243       
    82298244            sage: factor(-8*y - 4*x + z^2*(2*y + x)).factor_list()
    8230             [(z - 2, 1), (z + 2, 1), (x + 2*y, 1)]
     8245            [(x + 2*y, 1), (z + 2, 1), (z - 2, 1)]
    82318246
    82328247        A more complicated example::
    82338248       
     
    82358250            (x, u, v)
    82368251            sage: f = expand((2*u*v^2-v^2-4*u^3)^2 * (-u)^3 * (x-sin(x))^3)
    82378252            sage: f.factor()
    8238             -(x - sin(x))^3*(4*u^3 - 2*u*v^2 + v^2)^2*u^3
     8253            -(4*u^3 - 2*u*v^2 + v^2)^2*u^3*(x - sin(x))^3
    82398254            sage: g = f.factor_list(); g                     
    8240             [(x - sin(x), 3), (4*u^3 - 2*u*v^2 + v^2, 2), (u, 3), (-1, 1)]
     8255            [(4*u^3 - 2*u*v^2 + v^2, 2), (u, 3), (x - sin(x), 3), (-1, 1)]
    82418256
    82428257        This function also works for quotients::
    82438258       
     
    82458260            sage: g = f/(36*(1 + 2*y + y^2)); g
    82468261            1/36*(x^2*y^2 + 2*x*y^2 - x^2 + y^2 - 2*x - 1)/(y^2 + 2*y + 1)
    82478262            sage: g.factor(dontfactor=[x])
    8248             1/36*(y - 1)*(x^2 + 2*x + 1)/(y + 1)
     8263            1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1)
    82498264            sage: g.factor_list(dontfactor=[x])
    8250             [(y - 1, 1), (y + 1, -1), (x^2 + 2*x + 1, 1), (1/36, 1)]
     8265            [(x^2 + 2*x + 1, 1), (y + 1, -1), (y - 1, 1), (1/36, 1)]
    82518266                   
    82528267        This example also illustrates that the exponents do not have to be
    82538268        integers::
     
    82708285        EXAMPLES::
    82718286       
    82728287            sage: g = factor(x^3 - 1); g
    8273             (x - 1)*(x^2 + x + 1)
     8288            (x^2 + x + 1)*(x - 1)
    82748289            sage: v = g._factor_list(); v
    8275             [(x - 1, 1), (x^2 + x + 1, 1)]
     8290            [(x^2 + x + 1, 1), (x - 1, 1)]
    82768291            sage: type(v)
    82778292            <type 'list'>
    82788293        """
     
    84558470            sage: var('a,b,c,x')
    84568471            (a, b, c, x)
    84578472            sage: (a*x^2 + b*x + c).roots(x)
    8458             [(-1/2*(b + sqrt(-4*a*c + b^2))/a, 1), (-1/2*(b - sqrt(-4*a*c + b^2))/a, 1)]
     8473            [(-1/2*(b + sqrt(b^2 - 4*a*c))/a, 1), (-1/2*(b - sqrt(b^2 - 4*a*c))/a, 1)]
    84598474
    84608475        By default, all the roots are required to be explicit rather than
    84618476        implicit. To get implicit roots, pass ``explicit_solutions=False``
     
    84698484            ...
    84708485            RuntimeError: no explicit roots found
    84718486            sage: f.roots(explicit_solutions=False)
    8472             [((2^(8/9) - 2^(1/9) + x^(8/9) - x^(1/9))/(2^(8/9) - 2^(1/9)), 1)]
     8487            [((2^(8/9) + x^(8/9) - 2^(1/9) - x^(1/9))/(2^(8/9) - 2^(1/9)), 1)]
    84738488
    84748489        Another example, but involving a degree 5 poly whose roots don't
    84758490        get computed explicitly::
     
    85158530            (f6, f5, f4, x)
    85168531            sage: e=15*f6*x^2 + 5*f5*x + f4
    85178532            sage: res = e.roots(x); res
    8518             [(-1/30*(5*f5 + sqrt(-60*f4*f6 + 25*f5^2))/f6, 1), (-1/30*(5*f5 - sqrt(-60*f4*f6 + 25*f5^2))/f6, 1)]
     8533            [(-1/30*(5*f5 + sqrt(25*f5^2 - 60*f4*f6))/f6, 1), (-1/30*(5*f5 - sqrt(25*f5^2 - 60*f4*f6))/f6, 1)]
    85198534            sage: e.subs(x=res[0][0]).is_zero()
    85208535            True
    85218536        """
     
    89638978            sage: a.solve(t)
    89648979            []
    89658980            sage: b = a.simplify_radical(); b
    8966             -23040*(25.0*e^(900*t) - 2.0*e^(1800*t) - 32.0)*e^(-2400*t)
     8981            -23040*(-2.0*e^(1800*t) + 25.0*e^(900*t) - 32.0)*e^(-2400*t)
    89678982            sage: b.solve(t)
    89688983            []
    89698984            sage: b.solve(t, to_poly_solve=True)
     
    93439358        ::
    93449359
    93459360            sage: (k * binomial(n, k)).sum(k, 1, n)
    9346             n*2^(n - 1)
     9361            2^(n - 1)*n
    93479362
    93489363        ::
    93499364
     
    96129627            sage: f*(-2/3)
    96139628            -2/3*x - 2 < -2/3*y + 4/3
    96149629            sage: f*(-pi)
    9615             -(x + 3)*pi < -(y - 2)*pi
     9630            -pi*(x + 3) < -pi*(y - 2)
    96169631
    96179632        Since the direction of the inequality never changes when doing
    96189633        arithmetic with equations, you can multiply or divide the
  • sage/symbolic/expression_conversions.py

    diff --git a/sage/symbolic/expression_conversions.py b/sage/symbolic/expression_conversions.py
    a b  
    232232            sage: c.get_fake_div(-x)
    233233            FakeExpression([x], <built-in function neg>)
    234234            sage: c.get_fake_div((2*x^3+2*x-1)/((x-2)*(x+1)))
    235             FakeExpression([2*x^3 + 2*x - 1, FakeExpression([x - 2, x + 1], <built-in function mul>)], <built-in function div>)
     235            FakeExpression([2*x^3 + 2*x - 1, FakeExpression([x + 1, x - 2], <built-in function mul>)], <built-in function div>)
    236236
    237237        Check if #8056 is fixed, i.e., if numerator is 1.::
    238238
     
    14251425
    14261426        sage: f = (2*x^3+2*x-1)/((x-2)*(x+1))
    14271427        sage: f._fast_callable_(etb)
    1428         div(add(add(mul(ipow(v_0, 3), 2), mul(v_0, 2)), -1), mul(add(v_0, -2), add(v_0, 1)))
     1428        div(add(add(mul(ipow(v_0, 3), 2), mul(v_0, 2)), -1), mul(add(v_0, 1), add(v_0, -2)))
    14291429
    14301430    """
    14311431    return FastCallableConverter(ex, etb)()
  • sage/symbolic/function.pyx

    diff --git a/sage/symbolic/function.pyx b/sage/symbolic/function.pyx
    a b  
    10381038        Test pickling expressions with symbolic functions::
    10391039           
    10401040            sage: u = loads(dumps(foo(x)^2 + foo(y) + x^y)); u
    1041             x^y + foo(x)^2 + foo(y)
     1041            foo(x)^2 + x^y + foo(y)
    10421042            sage: u.subs(y=0)
    10431043            foo(x)^2 + foo(0) + 1
    10441044            sage: u.subs(y=0).n()
  • sage/symbolic/function_factory.py

    diff --git a/sage/symbolic/function_factory.py b/sage/symbolic/function_factory.py
    a b  
    185185        -b*sin(a)
    186186
    187187        sage: g.substitute_function(cr, (sin(x) + cos(x)).function(x))
    188         -(sin(a) - cos(a))*b
     188        b*(cos(a) - sin(a))
    189189
    190190    In Sage 4.0, basic arithmetic with unevaluated functions is no
    191191    longer supported::
  • sage/symbolic/getitem.pyx

    diff --git a/sage/symbolic/getitem.pyx b/sage/symbolic/getitem.pyx
    a b  
    6464
    6565        sage: x,y,z = var('x,y,z')
    6666        sage: e = x + x*y + z^y + 3*y*z; e
    67         x*y + 3*y*z + z^y + x
     67        x*y + 3*y*z + x + z^y
    6868        sage: e.op[1]
    6969        3*y*z
    7070        sage: e.op[1,1]
    7171        z
    7272        sage: e.op[-1]
    73         x
     73        z^y
    7474        sage: e.op[1:]
    75         [3*y*z, z^y, x]
     75        [3*y*z, x, z^y]
    7676        sage: e.op[:2]
    7777        [x*y, 3*y*z]
    7878        sage: e.op[-2:]
    79         [z^y, x]
     79        [x, z^y]
    8080        sage: e.op[:-2]
    8181        [x*y, 3*y*z]
    8282        sage: e.op[-5]
  • sage/symbolic/integration/integral.py

    diff --git a/sage/symbolic/integration/integral.py b/sage/symbolic/integration/integral.py
    a b  
    456456                 x y  + Sqrt[--] FresnelS[Sqrt[--] x]
    457457                             2                 Pi
    458458        sage: print f.integral(x)
    459         y^z*x + 1/8*((I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x) + (I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x))*sqrt(pi)
     459        x*y^z + 1/8*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x))
    460460
    461461    Alternatively, just use algorithm='mathematica_free' to integrate via Mathematica
    462462    over the internet (does NOT require a Mathematica license!)::
     
    489489    ::
    490490
    491491        sage: A = integral(1/ ((x-4) * (x^3+2*x+1)), x); A
    492         1/73*log(x - 4) - 1/73*integrate((x^2 + 4*x + 18)/(x^3 + 2*x + 1), x)
     492        -1/73*integrate((x^2 + 4*x + 18)/(x^3 + 2*x + 1), x) + 1/73*log(x - 4)
    493493
    494494    We now show that floats are not converted to rationals
    495495    automatically since we by default have keepfloat: true in maxima.
     
    519519
    520520        sage: assume(a>0)
    521521        sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
    522         2/9*sqrt(3)*b^2*arctan(1/3*(2*(b*x + a)^(1/3) + a^(1/3))*sqrt(3)/a^(1/3))/a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)
     522        2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)
    523523
    524524    TESTS:
    525525
     
    527527    see #3013::
    528528
    529529        sage: integrate(sin(x)*cos(10*x)*log(x), x)
    530         1/198*(11*cos(9*x) - 9*cos(11*x))*log(x) + 1/44*Ei(-11*I*x) - 1/36*Ei(-9*I*x) - 1/36*Ei(9*I*x) + 1/44*Ei(11*I*x)
     530        -1/198*(9*cos(11*x) - 11*cos(9*x))*log(x) + 1/44*Ei(11*I*x) - 1/36*Ei(9*I*x) - 1/36*Ei(-9*I*x) + 1/44*Ei(-11*I*x)
    531531
    532532    It is no longer possible to use certain functions without an
    533533    explicit variable.  Instead, evaluate the function at a variable,
     
    554554        Is  50015104*y^2-50015103  positive, negative, or zero?
    555555        sage: assume(y>1)
    556556        sage: res = integral(f,x,0.0001414, 1.); res
    557         2*y*arctan(1/y) - 2*y*arctan(0.0001414/y) - 0.0001414*log(y^2 + 1.999396e-08) + log(y^2 + 1.0) - 1.9997172
     557        -2*y*arctan(0.0001414/y) + 2*y*arctan(1/y) + log(y^2 + 1.0) - 0.0001414*log(y^2 + 1.999396e-08) - 1.9997172
    558558        sage: nres = numerical_integral(f.subs(y=2), 0.0001414, 1.); nres
    559559        (1.4638323264144..., 1.6251803529759...e-14)
    560560        sage: res.subs(y=2).n()
     
    592592        sage: integrate(t*cos(-theta*t),(t,-oo,oo))
    593593        integrate(t*cos(t*theta), t, -Infinity, +Infinity)
    594594
    595     Check if #6189 is fixed (which, by the way, also
    596     demonstrates it's not always good to expand)::
     595    Check if #6189 is fixed::
    597596
    598597        sage: n = N; n
    599598        <function numerical_approx at ...>
     
    603602        0.000000000000000
    604603        sage: integrate( ((F(x)-G(x))^2).expand(), x, -infinity, infinity).n()
    605604        -6.26376265908397e-17
    606         sage: integrate( (F(x)-G(x))^2, x, -infinity, infinity).n()
    607         -6.26376265908397e-17
     605        sage: integrate( (F(x)-G(x))^2, x, -infinity, infinity).n()# abstol 1e-6
     606        0
    608607
    609608    This was broken before Maxima 5.20::
    610609
     
    641640
    642641        sage: actual_result = integral(e^(-1/x^2), x, 0, 1)
    643642        sage: actual_result.full_simplify()
    644         ((e*erf(1) - e)*sqrt(pi) + 1)*e^(-1)
     643        (sqrt(pi)*(erf(1)*e - e) + 1)*e^(-1)
    645644        sage: ideal_result = 1/2*gamma(-1/2, 1)
    646645        sage: error = actual_result - ideal_result
    647646        sage: error.numerical_approx() # abs tol 1e-10
  • sage/symbolic/maxima_wrapper.py

    diff --git a/sage/symbolic/maxima_wrapper.py b/sage/symbolic/maxima_wrapper.py
    a b  
    1717        EXAMPLES::
    1818
    1919            sage: t = sin(x)^2 + cos(x)^2; t
    20             sin(x)^2 + cos(x)^2
     20            cos(x)^2 + sin(x)^2
    2121            sage: res = t.maxima_methods().trigsimp(); res
    2222            1
    2323            sage: type(res)
     
    3838        EXAMPLES::
    3939
    4040            sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
    41             log(sqrt(2) - 1) + log(sqrt(2) + 1)
     41            log(sqrt(2) + 1) + log(sqrt(2) - 1)
    4242            sage: u = t.maxima_methods(); u
    43             MaximaWrapper(log(sqrt(2) - 1) + log(sqrt(2) + 1))
     43            MaximaWrapper(log(sqrt(2) + 1) + log(sqrt(2) - 1))
    4444            sage: type(u)
    4545            <class 'sage.symbolic.maxima_wrapper.MaximaWrapper'>
    4646            sage: u.logcontract()
    47             log((sqrt(2) - 1)*(sqrt(2) + 1))
     47            log((sqrt(2) + 1)*(sqrt(2) - 1))
    4848            sage: u.logcontract().parent()
    4949            Symbolic Ring
    5050
     
    7070        EXAMPLES::
    7171
    7272            sage: t = sin(x)^2 + cos(x)^2; t
    73             sin(x)^2 + cos(x)^2
     73            cos(x)^2 + sin(x)^2
    7474            sage: u = t.maxima_methods()
    7575            sage: import sagenb.misc.support as s
    7676            sage: s.completions('u.airy_',globals(),system='python')
     
    7878            sage: type(u.airy_ai)
    7979            <class 'sage.symbolic.maxima_wrapper.MaximaFunctionElementWrapper'>
    8080            sage: u.airy_ai()
    81             airy_ai(sin(x)^2 + cos(x)^2)
     81            airy_ai(cos(x)^2 + sin(x)^2)
    8282        """
    8383        if self._maxima_exp is None:
    8484            self._maxima_exp = self._exp._maxima_()
     
    9696        EXAMPLES::
    9797
    9898            sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
    99             log(sqrt(2) - 1) + log(sqrt(2) + 1)
     99            log(sqrt(2) + 1) + log(sqrt(2) - 1)
    100100            sage: u = t.maxima_methods().sage()
    101101            sage: u is t
    102102            True
     
    108108        EXAMPLES::
    109109
    110110            sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
    111             log(sqrt(2) - 1) + log(sqrt(2) + 1)
     111            log(sqrt(2) + 1) + log(sqrt(2) - 1)
    112112            sage: u = t.maxima_methods()
    113113            sage: SR(u) is t # indirect doctest
    114114            True
     
    120120        EXAMPLES::
    121121
    122122            sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
    123             log(sqrt(2) - 1) + log(sqrt(2) + 1)
     123            log(sqrt(2) + 1) + log(sqrt(2) - 1)
    124124            sage: u = t.maxima_methods(); u
    125             MaximaWrapper(log(sqrt(2) - 1) + log(sqrt(2) + 1))
     125            MaximaWrapper(log(sqrt(2) + 1) + log(sqrt(2) - 1))
    126126            sage: loads(dumps(u))
    127             MaximaWrapper(log(sqrt(2) - 1) + log(sqrt(2) + 1))
     127            MaximaWrapper(log(sqrt(2) + 1) + log(sqrt(2) - 1))
    128128        """
    129129        return (MaximaWrapper, (self._exp,))
    130130
     
    133133        EXAMPLES::
    134134
    135135            sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
    136             log(sqrt(2) - 1) + log(sqrt(2) + 1)
     136            log(sqrt(2) + 1) + log(sqrt(2) - 1)
    137137            sage: u = t.maxima_methods(); u
    138             MaximaWrapper(log(sqrt(2) - 1) + log(sqrt(2) + 1))
     138            MaximaWrapper(log(sqrt(2) + 1) + log(sqrt(2) - 1))
    139139            sage: u._repr_()
    140             'MaximaWrapper(log(sqrt(2) - 1) + log(sqrt(2) + 1))'
     140            'MaximaWrapper(log(sqrt(2) + 1) + log(sqrt(2) - 1))'
    141141        """
    142142        return "MaximaWrapper(%s)"%(self._exp)
  • sage/symbolic/random_tests.py

    diff --git a/sage/symbolic/random_tests.py b/sage/symbolic/random_tests.py
    a b  
    335335        sage: for i,j in CartesianProduct(range(0,3),range(0,3)):
    336336        ...       cmp[i,j] = x[i].__cmp__(x[j])
    337337        sage: cmp
    338         [ 0  1  1]
    339         [-1  0 -1]
    340         [-1  1  0]
     338        [ 0 -1 -1]
     339        [ 1  0 -1]
     340        [ 1  1  0]
    341341    """
    342342    from sage.matrix.constructor import matrix
    343343    from sage.combinat.cartesian_product import CartesianProduct
  • sage/symbolic/relation.py

    diff --git a/sage/symbolic/relation.py b/sage/symbolic/relation.py
    a b  
    1010    a*x^2 + b*x + c == 0
    1111    sage: print solve(qe, x)
    1212    [
    13     x == -1/2*(b + sqrt(-4*a*c + b^2))/a,
    14     x == -1/2*(b - sqrt(-4*a*c + b^2))/a
     13    x == -1/2*(b + sqrt(b^2 - 4*a*c))/a,
     14    x == -1/2*(b - sqrt(b^2 - 4*a*c))/a
    1515    ]
    1616
    1717
     
    238238    sage: var('x,y,z,w')
    239239    (x, y, z, w)
    240240    sage: f =  (x+y+w) == (x^2 - y^2 - z^3);   f
    241     w + x + y == x^2 - y^2 - z^3
     241    w + x + y == -z^3 + x^2 - y^2
    242242    sage: f.variables()
    243243    (w, x, y, z)
    244244
     
    276276    x == y - 5
    277277    sage: h =  x^3 + sqrt(2) == x*y*sin(x)
    278278    sage: h
    279     sqrt(2) + x^3 == x*y*sin(x)
     279    x^3 + sqrt(2) == x*y*sin(x)
    280280    sage: h - sqrt(2)
    281281    x^3 == x*y*sin(x) - sqrt(2)
    282282    sage: h + f
    283     x + sqrt(2) + x^3 + 3 == x*y*sin(x) + y - 2
     283    x^3 + x + sqrt(2) + 3 == x*y*sin(x) + y - 2
    284284    sage: f = x + 3 < y - 2
    285285    sage: g = 2 < x+10
    286286    sage: f - g
     
    648648
    649649       sage: x,y=var('x y'); c1(x,y)=(x-5)^2+y^2-16; c2(x,y)=(y-3)^2+x^2-9
    650650       sage: solve([c1(x,y),c2(x,y)],[x,y])                               
    651        [[x == -9/68*sqrt(55) + 135/68, y == -15/68*sqrt(5)*sqrt(11) + 123/68], [x == 9/68*sqrt(55) + 135/68, y == 15/68*sqrt(5)*sqrt(11) + 123/68]]
     651       [[x == -9/68*sqrt(55) + 135/68, y == -15/68*sqrt(11)*sqrt(5) + 123/68], [x == 9/68*sqrt(55) + 135/68, y == 15/68*sqrt(11)*sqrt(5) + 123/68]]
    652652       
    653653    TESTS::
    654654
  • sage/symbolic/ring.pyx

    diff --git a/sage/symbolic/ring.pyx b/sage/symbolic/ring.pyx
    a b  
    318318            sage: x,y = var('x,y')
    319319            sage: w0 = SR.wild(0); w1 = SR.wild(1)
    320320            sage: pattern = sin(x)*w0*w1^2; pattern
    321             $0*$1^2*sin(x)
     321            $1^2*$0*sin(x)
    322322            sage: f = atan(sin(x)*3*x^2); f
    323323            arctan(3*x^2*sin(x))
    324324            sage: f.has(pattern)
  • sage/symbolic/units.py

    diff --git a/sage/symbolic/units.py b/sage/symbolic/units.py
    a b  
    12781278    You can also convert quantities of units::
    12791279
    12801280        sage: sage.symbolic.units.convert(cos(50) * units.angles.radian, units.angles.degree)
    1281         (180*cos(50)/pi)*degree
     1281        degree*(180*cos(50)/pi)
    12821282        sage: sage.symbolic.units.convert(cos(30) * units.angles.radian, units.angles.degree).polynomial(RR)
    12831283        8.83795706233228*degree
    12841284        sage: sage.symbolic.units.convert(50 * units.length.light_year / units.time.year, units.length.foot / units.time.second)
     
    12871287    Quantities may contain variables (not for temperature conversion, though)::
    12881288
    12891289        sage: sage.symbolic.units.convert(50 * x * units.area.square_meter, units.area.acre)
    1290         (1953125/158080329*x)*acre
     1290        acre*(1953125/158080329*x)
    12911291    """
    12921292    base_target = target
    12931293    z = {}
  • sage/tensor/differential_form_element.py

    diff --git a/sage/tensor/differential_form_element.py b/sage/tensor/differential_form_element.py
    a b  
    13101310        sage: g[1] = sin(y); g
    13111311        sin(y)*dy
    13121312        sage: wedge(f, g)
    1313         -sin(y)*cos(x)*dy/\dz
     1313        -cos(x)*sin(y)*dy/\dz
    13141314        sage: f.wedge(g)
    1315         -sin(y)*cos(x)*dy/\dz
     1315        -cos(x)*sin(y)*dy/\dz
    13161316        sage: wedge(f, g) == f.wedge(g)
    13171317        True
    13181318    """
  • sage/tests/french_book/polynomes.py

    diff --git a/sage/tests/french_book/polynomes.py b/sage/tests/french_book/polynomes.py
    a b  
    77
    88  sage: x = var('x'); p = (2*x+1)*(x+2)*(x^4-1)
    99  sage: print p, "est de degré", p.degree(x)
    10   (x + 2)*(2*x + 1)*(x^4 - 1) est de degré 6
     10  (x^4 - 1)*(2*x + 1)*(x + 2) est de degré 6
    1111
    1212Sage example in ./polynomes.tex, line 69::
    1313
  • sage/tests/french_book/recequadiff.py

    diff --git a/sage/tests/french_book/recequadiff.py b/sage/tests/french_book/recequadiff.py
    a b  
    6868
    6969  sage: DE = diff(y,x)+2*y == x**2-2*x+3
    7070  sage: desolve(DE, y)
    71   -1/4*(2*(2*x - 1)*e^(2*x) - (2*x^2 - 2*x + 1)*e^(2*x) - 4*c
    72   - 6*e^(2*x))*e^(-2*x)
     71  1/4*((2*x^2 - 2*x + 1)*e^(2*x) - 2*(2*x - 1)*e^(2*x) + 4*c
     72  + 6*e^(2*x))*e^(-2*x)
    7373
    7474Sage example in ./recequadiff.tex, line 305::
    7575
    7676  sage: desolve(DE, y).expand()
    77   c*e^(-2*x) + 1/2*x^2 - 3/2*x + 9/4
     77  1/2*x^2 + c*e^(-2*x) - 3/2*x + 9/4
    7878
    7979Sage example in ./recequadiff.tex, line 321::
    8080
     
    190190Sage example in ./recequadiff.tex, line 575::
    191191
    192192  sage: Sol(x) = solve(sol, y)[0]; Sol(x)
    193   log(y(x)) == (c + x)*a + log(b*y(x) - a)
     193  log(y(x)) == a*(c + x) + log(b*y(x) - a)
    194194
    195195Sage example in ./recequadiff.tex, line 582::
    196196
    197197  sage: Sol(x) = Sol(x).lhs()-Sol(x).rhs(); Sol(x)
    198   -(c + x)*a - log(b*y(x) - a) + log(y(x))
     198  -a*(c + x) - log(b*y(x) - a) + log(y(x))
    199199  sage: Sol = Sol.simplify_log(); Sol(x)
    200   -(c + x)*a + log(y(x)/(b*y(x) - a))
     200  -a*(c + x) + log(y(x)/(b*y(x) - a))
    201201  sage: solve(Sol, y)[0].simplify()
    202202  y(x) == a*e^(a*c + a*x)/(b*e^(a*c + a*x) - 1)
    203203
     
    206206  sage: x = var('x'); y = function('y', x)
    207207  sage: DE = diff(y,x,2)+3*y == x^2-7*x+31
    208208  sage: desolve(DE, y).expand()
    209   k1*sin(sqrt(3)*x) + k2*cos(sqrt(3)*x) + 1/3*x^2 - 7/3*x + 91/9
     209  1/3*x^2 + k2*cos(sqrt(3)*x) + k1*sin(sqrt(3)*x) - 7/3*x + 91/9
    210210
    211211Sage example in ./recequadiff.tex, line 611::
    212212
     
    217217Sage example in ./recequadiff.tex, line 621::
    218218
    219219  sage: desolve(DE, y, ics=[0,1,-1,0]).expand()
    220   1/3*x^2 - 7/3*x - 82/9*sin(sqrt(3)*x)*cos(sqrt(3))/sin(sqrt(3))
     220  1/3*x^2 - 7/3*x - 82/9*cos(sqrt(3))*sin(sqrt(3)*x)/sin(sqrt(3))
    221221  + 115/9*sin(sqrt(3)*x)/sin(sqrt(3)) - 82/9*cos(sqrt(3)*x) + 91/9
    222222
    223223Sage example in ./recequadiff.tex, line 674::
     
    265265
    266266  sage: X(s) = 1/(s^2-3*s-4)/(s^2+1) + (s-4)/(s^2-3*s-4)
    267267  sage: X(s).inverse_laplace(s, x)
    268   9/10*e^(-x) + 1/85*e^(4*x) - 5/34*sin(x) + 3/34*cos(x)
     268  3/34*cos(x) + 1/85*e^(4*x) + 9/10*e^(-x) - 5/34*sin(x)
    269269
    270270Sage example in ./recequadiff.tex, line 807::
    271271
    272272  sage: X(s).partial_fraction()
    273   1/34*(3*s - 5)/(s^2 + 1) + 1/85/(s - 4) + 9/10/(s + 1)
     273  1/34*(3*s - 5)/(s^2 + 1) + 9/10/(s + 1) + 1/85/(s - 4)
    274274
    275275Sage example in ./recequadiff.tex, line 818::
    276276
    277277  sage: x = var('x'); y = function('y',x)
    278278  sage: eq = diff(y,x,x) - 3*diff(y,x) - 4*y - sin(x) == 0
    279279  sage: desolve_laplace(eq, y)
    280   1/10*(8*y(0) - 2*D[0](y)(0) - 1)*e^(-x) + 1/85*(17*y(0) +
    281   17*D[0](y)(0) + 1)*e^(4*x) - 5/34*sin(x) + 3/34*cos(x)
     280  1/85*(17*y(0) + 17*D[0](y)(0) + 1)*e^(4*x) + 1/10*(8*y(0)
     281  - 2*D[0](y)(0) - 1)*e^(-x) + 3/34*cos(x) - 5/34*sin(x)
    282282  sage: desolve_laplace(eq, y, ics=[0,1,-1])
    283   9/10*e^(-x) + 1/85*e^(4*x) - 5/34*sin(x) + 3/34*cos(x)
     283  3/34*cos(x) + 1/85*e^(4*x) + 9/10*e^(-x) - 5/34*sin(x)
    284284
    285285Sage example in ./recequadiff.tex, line 869::
    286286
     
    290290  sage: A = matrix([[2,-2,0],[-2,0,2],[0,2,2]])
    291291  sage: system = [diff(y[i], x) - (A * y)[i] for i in range(3)]
    292292  sage: desolve_system(system, [y1, y2, y3], ics=[0,2,1,-2])
    293   [y1(x) ==   e^(-2*x) + e^(4*x),
    294    y2(x) == 2*e^(-2*x) - e^(4*x),
    295    y3(x) ==  -e^(-2*x) - e^(4*x)]
     293  [y1(x) == e^(4*x) + e^(-2*x),
     294   y2(x) == -e^(4*x) + 2*e^(-2*x),
     295   y3(x) == -e^(4*x) - e^(-2*x)]
    296296
    297297Sage example in ./recequadiff.tex, line 913::
    298298
     
    301301  sage: A = matrix([[3,-4],[1,3]])
    302302  sage: system = [diff(y[i], x) - (A * y)[i] for i in range(2)]
    303303  sage: desolve_system(system, [y1, y2], ics=[0,2,0])
    304   [y1(x) == 2*e^(3*x)*cos(2*x), y2(x) == e^(3*x)*sin(2*x)]
     304  [y1(x) == 2*cos(2*x)*e^(3*x), y2(x) == e^(3*x)*sin(2*x)]
    305305
    306306Sage example in ./recequadiff.tex, line 966::
    307307
     
    315315Sage example in ./recequadiff.tex, line 977::
    316316
    317317  sage: sol[0]
    318   u1(x) == 1/24*(2*u1(0) - 6*u2(0) - u3(0) + 3*u4(0))*e^(-4*x)
    319           + 1/12*(2*u1(0) - 6*u2(0) + 5*u3(0) + 3*u4(0))*e^(2*x)
    320           + 3/4*u1(0) + 3/4*u2(0) - 3/8*u3(0) - 3/8*u4(0)
     318  u1(x) == 1/12*(2*u1(0) - 6*u2(0) + 5*u3(0) + 3*u4(0))*e^(2*x)
     319           + 1/24*(2*u1(0) - 6*u2(0) - u3(0) + 3*u4(0))*e^(-4*x)
     320           + 3/4*u1(0) + 3/4*u2(0) - 3/8*u3(0) - 3/8*u4(0)
     321
    321322  sage: sol[1]
    322323  u2(x) == -1/12*(2*u1(0) - 6*u2(0) - u3(0) - 3*u4(0))*e^(2*x)
    323324           - 1/24*(2*u1(0) - 6*u2(0) - u3(0) + 3*u4(0))*e^(-4*x)