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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.
>        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: 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 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

> `

## 13.7 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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