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13. specfun_ct.ct

This section describes functions from file specfun_ct.ct.

13.1 ChebyshevT

[w] = ChebyshevT(n,x)
 ChebyshevT(n,x) gives the nth Chebyshev polynomial of the first kind,
   ChebyshevT(n,cos(theta)):= cos(n*theta).
   Error codes:
   -1: First arg not numerical
   -2: Second arg not numerical
   -5: Argument dimensions do not match
 

13.2 Gauss2F1

[w] = Gauss2F1(a,b,c,z)
 Gauss2F1(a,b,c,z) is Gauss' 2F1 hypergeometric function
   F(a,b;c;z), defined by the power series
      F(a,b;c;z) = sum(((a)_n*(b)*n/(c)_n)*z^n/n!,n=0..Inf)
   where (a)_n is the Pochhammer symbol poch(a,n),
      (a)_n = a*(a+1)*...*(a+n-1) = Gamma(a+n)/Gamma(a).
   The function has a singularity at z=1 and a branch cut
   in Re(z) > 1, Im(z) == 0. The function is evaluated for
   any numerical arguments and is threaded automatically over
   array arguments. The method of evaluation is power series
   expansion for small |z|, a transformation formula for certain
   large |z|, and direct integration of the defining diff. equation
   in other cases.
See also: Kummer1F1.
   Error codes:
   -1: Args must be complex scalars
   -2: Second arg not numerical
   -3: Third arg not numerical
   -4: Fourth arg not numerical
   -5: Argument dimensions do not match

13.3 IncompleteGammaP

[w] = IncompleteGammaP(a,x)
 IncompleteGammaP(a,x) is the regularized incomplete gamma function
     IncompleteGammaP(a,x) := integrate(exp(-t)*t^(a-1),t=0..x)/Gamma(a).
   The evaluation is reliable for Re(a) > 0 only, although a result
   is returned also for Re(a) <= 0.
   The notation conforms with Numerical Recipes.
   Error codes:
   -1: First arg not numerical
   -2: Second arg not numerical
   -5: Argument dimensions do not match
 

13.4 IncompleteGammaQ

[w] = IncompleteGammaQ(a,x)
 IncompleteGammaQ(a,x) is the regularized incomplete gamma function
     IncompleteGammaQ(a,x) := 1 - IncompleteGammaP(a,x).
   The evaluation is reliable for Re(a) > 0 only, although a result
   is returned also for Re(a) <= 0.
   The notation conforms with Numerical Recipes.
   Error codes:
   -1: First arg not numerical
   -2: Second arg not numerical
   -5: Argument dimensions do not match
 

13.5 Kummer1F1

[w] = Kummer1F1(a,c,z)
 Kummer1F1(a,c,z) is the 1F1 confluent hypergeometric function,
   defined by
     1F1(a,c,z) = sum(((a)_n/(c)_n)*z^n/n!,n=0..Inf)
   where (a)_n is the Pochhammer symbol poch(a,n),
      (a)_n = a*(a+1)*...*(a+n-1) = Gamma(a+n)/Gamma(a).
   The function has no singularities or branch cuts for any z,
   but is infinite when c is a non-positive integer.
   The method of evaluation is power series expansion for
   small and moderate |z| and direct integration of the defining
   diff. equation in other cases.
See also: Kummer1F1Regularized, Gauss2F1.
   Error codes:
   -1: First arg not numerical
   -2: Second arg not numerical
   -3: Third arg not numerical
   -5: Argument dimensions do not match

13.6 Kummer1F1Regularized

[w] = Kummer1F1Regularized(a,c,z)
 Kummer1F1Regularized(a,c,z) is the regularized confluent
   hypergeometric function, defined by
   Kummer1F1Regularized(a,c,z):= Kummer1F1(a,c,z)/Gamma(c).
   This function is finite for all finite values of the parameters.
   The evaluation method works also when c is negative integer.
See also: Kummer1F1.
   Error codes:
   -1: First arg not numerical
   -2: Second arg not numerical
   -3: Third arg not numerical
   -5: Argument dimensions do not match

13.7 LegendreP

[w] = LegendreP(n,m;x)
 LegendreP(n,x) is the nth Legendre polynomial P_n(x)
   evaluated at x.
   LegendreP(n,m,x) is the associated Legendre function
   P_n^m(x).
   The Arfken definition is in use, which is different from
   e.g. what Mathematica uses.
   Error codes:
   -1: First arg not numerical
   -2: Second arg not numerical
   -3: Third arg not numerical
   -5: Argument dimensions do not match

13.8 logfact

[y] = logfact(x)
 logfact(x) returns log(x!) for any numerical x, including complex.
   The result for real or int x is usually real, but may be complex
   if x is negative. For arrays, the function is applied componentwise.
   In the array case, the result is complex if any of the imaginary
   parts becomes nonzero, otherwise real.
See also: <@@ref>factfact, <@@ref>factfact<@@ref>GammaGamma, <@@ref>factfact<@@ref>GammaGamma<@@ref>logGammalogGamma.
   Error codes:
   -1: Argument not numerical
 


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