# [Maxima] Implementation of Gamma functions

Raymond Toy toy.raymond at gmail.com
Fri Sep 5 23:55:07 CDT 2008

```On Fri, Sep 5, 2008 at 5:56 PM, Dieter Kaiser <drdieterkaiser at web.de> wrote:

> I have started to implement support for the Incomplete Gamma function.
> Maxima
> only knows the symbols gammaincomplete and gammagreek.
>
> I have already finished the routines for the numerical evaluation of the
> Incomplete Gamma function for Float, Complex float and Bigfloat. Now I am
> working on the routines for Complex Bigfloats. I have used again an
> expansion in
> a series or continued fractions for the Regularized Incomplete Gamma
> function.
>
Very cool!

>
> Or we could use names like
>
> gamma_incomplete(a,z)
> gamma_greek(a,z)                 = 1 - gamma_incomplete(a,z)
> gamma_incomplete_gen(a,z1,z2)    =
> gamma_incomplete(a,z1)-gamma_incomplete(a,z2)
> gamma_incomplete_reg(a,z)        = gamma_incomplete(a,z)/gamma(a)
> gamma_incomplete_gen_reg(a,z1,z2)= gamma_incomplete_gen(a,z1,z2)/gamma(a)
>
>
I prefer these names.  I find incomple and greek rather confusing because I
can never remember which one is the tail.

Maybe gamma_incomplete_tail for the integral from z to inf and
gamma_incomplete for the integral from 0 to z?

>
> Perhaps more interessting would be to implement a Log Gamma function and
> the
> Inverse Gamma functions.
>

Well, gamma_incomplete gives us a way to evaluate erf, which we don't have
today for complex or bigfloat numbers.

Since we have a gamma function, the log gamma isn't too hard for numerical
evaluation.  Inverse gamma is probably not too hard if a Newton iteration
can be applied.

Ray
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