This section describes functions from file specfun_ct.ct.
[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
[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
[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
[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
[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
[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
[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
[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