Ticket #11948: 11948.patch

sage/functions/other.py

@@ -28 +28 @@
     _eval_ = BuiltinFunction._eval_default
     def __init__(self):
         r"""
-        The error function, defined as
+        The error function, defined for real values as
         `\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt`.
+        This function is also defined for complex values, via analytic
+        continuation.
 
-        Sage currently only implements the error function (via a call to
-        PARI) when the input is real.
+        Sage implements the error function via the ``erfc()`` function in PARI.
 
         EXAMPLES::
 
@@ -43 +44 @@
             sage: loads(dumps(erf))
             erf
 
-        The following fails because we haven't implemented
-        erf yet for complex values::
-        
-            sage: complex(erf(3*I))
-            Traceback (most recent call last):
-            ...
-            TypeError: unable to simplify to complex approximation
-
         TESTS:
 
         Check if conversion from maxima elements work::
@@ -61 +54 @@
         """
         BuiltinFunction.__init__(self, "erf", latex_name=r"\text{erf}")
 
-    def _evalf_(self, x, parent=None):
+    def _evalf_(self, x, parent):
         """
         EXAMPLES::
 
             sage: erf(2).n()
             0.995322265018953
-            sage: erf(2).n(150)
-            Traceback (most recent call last):
-            ...
-            NotImplementedError: erf not implemented for precision higher than 53
+            sage: erf(2).n(200)
+            0.99532226501895273416206925636725292861089179704006007673835
+            sage: erf(pi - 1/2*I).n(100)
+            1.0000111669099367825726058952 + 1.6332655417638522934072124548e-6*I
+
+        TESTS:
+
+        Check that PARI/GP through the GP interface gives the same answer::
+
+            sage: gp.set_real_precision(59)  # random
+            38
+            sage: print gp.eval("1 - erfc(1)"); print erf(1).n(200);
+            0.84270079294971486934122063508260925929606699796630290845994
+            0.84270079294971486934122063508260925929606699796630290845994
         """
         try:
             prec = parent.prec()
         except AttributeError: # not a Sage parent
             prec = 0
-        if prec > 53:
-            raise NotImplementedError, "erf not implemented for precision higher than 53"
-        return parent(1 - pari(float(x)).erfc())
+        return parent(1) - parent(pari(x).erfc(prec))
         
     def _derivative_(self, x, diff_param=None):
         """

sage/symbolic/expression.pyx

@@ -988 +988 @@
         EXAMPLES::
 
             sage: complex(I)
-            1j
+            1j
             sage: complex(erf(3*I))
-            Traceback (most recent call last):
-            ...
-            TypeError: unable to simplify to complex approximation
+            (1.0000000000000002+1629.8673238578601j)
         """
         try:
             return self._eval_self(complex)
@@ -2927 +2925 @@
             sage: taylor(a*log(z), z, 2, 3)
             1/24*(z - 2)^3*a - 1/8*(z - 2)^2*a + 1/2*(z - 2)*a + a*log(2)
 
-        ::
+        ::
 
             sage: taylor(sqrt (sin(x) + a*x + 1), x, 0, 3)
             1/48*(3*a^3 + 9*a^2 + 9*a - 1)*x^3 - 1/8*(a^2 + 2*a + 1)*x^2 + 1/2*(a + 1)*x + 1
 
-        ::
+        ::
 
             sage: taylor (sqrt (x + 1), x, 0, 5)
             7/256*x^5 - 5/128*x^4 + 1/16*x^3 - 1/8*x^2 + 1/2*x + 1
 
-        ::
+        ::
 
             sage: taylor (1/log (x + 1), x, 0, 3)
             -19/720*x^3 + 1/24*x^2 - 1/12*x + 1/x + 1/2
 
-        ::
+        ::
 
             sage: taylor (cos(x) - sec(x), x, 0, 5)
             -1/6*x^4 - x^2
 
-        ::
+        ::
 
             sage: taylor ((cos(x) - sec(x))^3, x, 0, 9)
             -1/2*x^8 - x^6
 
-        ::
+        ::
 
             sage: taylor (1/(cos(x) - sec(x))^3, x, 0, 5)
             -15377/7983360*x^4 - 6767/604800*x^2 + 11/120/x^2 + 1/2/x^4 - 1/x^6 - 347/15120
 
- 
-        Ticket #7472 fixed (Taylor polynomial in more variables) ::
+        TESTS:
+
+        Check that ticket #7472 is fixed (Taylor polynomial in more variables)::
  
             sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,1),4) 
             (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
@@ -7548 +7547 @@
         -  ``multiplicities`` - bool (default: False); if True,
            return corresponding multiplicities.  This keyword is
            incompatible with ``to_poly_solve=True`` and does not make
-           any sense when solving inequality.
+           any sense when solving inequality.
         
         -  ``solution_dict`` - bool (default: False); if True or non-zero,
            return a list of dictionaries containing solutions. Not used
@@ -7562 +7561 @@
            Maxima's ``to_poly_solver`` package to search for more possible 
            solutions, but possibly encounter approximate solutions.
            This keyword is incompatible with ``multiplicities=True``
-           and is not used when solving inequality. Setting ``to_poly_solve``
+           and is not used when solving inequality. Setting ``to_poly_solve``
            to 'force' (string) omits Maxima's solve command (usefull when
            some solution of trigonometric equations are lost).
         
@@ -7610 +7609 @@
             sage: solve(Q*sqrt(Q^2 + 2) - 1, Q)
             [Q == 1/sqrt(Q^2 + 2)]
             sage: solve(Q*sqrt(Q^2 + 2) - 1, Q, to_poly_solve=True)
-                        [Q == 1/sqrt(-sqrt(2) + 1), Q == 1/sqrt(sqrt(2) + 1)]
+            [Q == 1/sqrt(-sqrt(2) + 1), Q == 1/sqrt(sqrt(2) + 1)]
 
         In some cases there may be infinitely many solutions indexed
         by a dummy variable.  If it begins with ``z``, it is implicitly