# HG changeset patch
# User Stefan Reiterer <domors@gmx.net>
# Date 1386541142 3600
# Node ID 738e19cfa1802702b6f6e8c501efacd9022eb631
# Parent 76e0bcf8d3bc159779269cb823abf4efd4bbfc33
trac 9706: corrections in documentation, and references
diff git a/sage/functions/orthogonal_polys.py b/sage/functions/orthogonal_polys.py
a

b


270  270  
271  271  REFERENCES: 
272  272  
273    [ASHandbook] Abramowitz and Stegun: Handbook of Mathematical Functions, 
274   http://www.math.sfu.ca/ cbm/aands/ 
 273  .. [ASHandbook] Abramowitz and Stegun: Handbook of Mathematical Functions, 
 274  http://www.math.sfu.ca/ cbm/aands/ 
275  275  
276    :wikipedia:`Chebyshev_polynomials` 
 276  .. :wikipedia:`Chebyshev_polynomials` 
277  277  
278    :wikipedia:`Legendre_polynomials` 
 278  .. :wikipedia:`Legendre_polynomials` 
279  279  
280    :wikipedia:`Hermite_polynomials` 
 280  .. :wikipedia:`Hermite_polynomials` 
281  281  
282    http://mathworld.wolfram.com/GegenbauerPolynomial.html 
 282  .. http://mathworld.wolfram.com/GegenbauerPolynomial.html 
283  283  
284    :wikipedia:`Jacobi_polynomials` 
 284  .. :wikipedia:`Jacobi_polynomials` 
285  285  
286    :wikipedia:`Laguerre_polynomia` 
 286  .. :wikipedia:`Laguerre_polynomia` 
287  287  
288    :wikipedia:`Associated_Legendre_polynomials` 
 288  .. :wikipedia:`Associated_Legendre_polynomials` 
289  289  
290    [EffCheby] Wolfram Koepf: Effcient Computation of Chebyshev Polynomials 
291   in Computer Algebra 
292   Computer Algebra Systems: A Practical Guide. 
293   John Wiley, Chichester (1999): 7999. 
 290  .. [EffCheby] Wolfram Koepf: Effcient Computation of Chebyshev Polynomials 
 291  in Computer Algebra 
 292  Computer Algebra Systems: A Practical Guide. 
 293  John Wiley, Chichester (1999): 7999. 
294  294  
295  295  AUTHORS: 
296  296  
… 
… 

424  424  
425  425  def _apply_formula_(self, *args): 
426  426  """ 
427   The Clenshaw method uses the three term recursion of the polynomial, 
 427  Method which uses the three term recursion of the polynomial, 
428  428  or explicit formulas instead of maxima to evaluate the polynomial 
429  429  efficiently, if the `x` argument is not a symbolic expression. 
430   The name comes from the Clenshaw algorithm for fast evaluation of 
431   polynomials in chebyshev form. 
432  430  
433  431  EXAMPLES:: 
434  432  
… 
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456  454  
457  455  def _eval_(self, *args): 
458  456  """ 
459   The symbolic evaluation is either done with maxima, or with direct 
460   computaion in pynac. 
461   
462   For the fast numerical evaluation, another method should be used. 
463   We use Clenshaw's algorithm, which uses the recursion. 
464   The function also checks for special values, and if 
465   the order is an integer and in range. 
 457  The _eval_ method decides which evaluation suits best 
 458  for the given input, and returns a proper value. 
466  459  
467  460  EXAMPLES:: 
468  461  
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736  729  
737  730  def _apply_formula_(self,*args): 
738  731  """ 
739   Clenshaw method for :class:`chebyshev_T` (means use recursions in this 
740   case). This is much faster for numerical evaluation than maxima! 
 732  Applies explicit formulas for :class:`chebyshev_T`. 
 733  This is much faster for numerical evaluation than maxima! 
741  734  See [ASHandbook]_ 227 (p. 782) for details for the recurions. 
742   See also [EffCheby] for fast evaluation techniques. 
 735  See also [EffCheby]_ for fast evaluation techniques. 
743  736  
744  737  EXAMPLES:: 
745  738  
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758  751  return 1 
759  752  if k == 1: 
760  753  return x 
761   
762   # TODO: When evaluation of Symbolic Expressions works better 
763   # use these explicit formulas instead! 
764   #if 1 <= x <= 1: 
765   # return cos(k*acos(x)) 
766   #elif 1 < x: 
767   # return cosh(k*acosh(x)) 
768   #else: # x < 1 
769   # return (1)**(k%2)*cosh(k*acosh(x)) 
770  754  
771  755  help1 = 1 
772  756  help2 = x 
… 
… 

885  869  
886  870  def _apply_formula_(self,*args): 
887  871  """ 
888   Clenshaw method for :class:`chebyshev_U` (means we use the recursion) 
 872  Applies explicit formulas for :class:`chebyshev_U`. 
889  873  This is much faster for numerical evaluation than maxima. 
890  874  See [ASHandbook]_ 227 (p. 782) for details on the recurions. 
891   See also [ 
 875  See also [EffCheby]_ for the recursion formulas. 
892  876  
893  877  EXAMPLES:: 
894  878  
… 
… 

1113  1097  Returns the generalized Laguerre polynomial for integers `n > 1`. 
1114  1098  Typically, `a = 1/2` or `a = 1/2`. 
1115  1099  
1116   REFERENCE: 
 1100  REFERENCES: 
1117  1101  
1118  1102   Table on page 789 in [ASHandbook]_. 
1119  1103  