# [Maxima] gamma_incomplete(1/2,0)?

Dieter Kaiser drdieterkaiser at web.de
Tue Dec 1 17:07:40 CST 2009

```Am Dienstag, den 01.12.2009, 09:27 -0500 schrieb Raymond Toy:
> Dieter Kaiser wrote:
> > Am Montag, den 30.11.2009, 19:03 -0500 schrieb Raymond Toy:
> >
> >> Current CVS says gamma_incomplete(1/2,0) is sqrt(%pi)/2.  But the help
> >> for gamma_incomplete says gamma_incomplete(a,z) =
> >> integrate(exp(-t)*t^(a-1),t,0,z).  (This matches A&S 6.5.2.)
> >>
> >> So from the documentation, I would expect the integral to be 0.
> >>
> >> Which is correct?  Since A&S is our reference, I think the current value
> >> is incorrect and we should return 0.
> >>
> >
> > Sorry, but I think the documentation is wrong. The problem is that the
> > Incomplete Gamma function and the lower Incomplete Gamma function are
> > easy to mix up. I think these are the defintions (e.g. wolfram.com)
> >
> > Incomplete Gamma functions:
> >
> >  gamma_incomplete(a,z) = integrate(exp(-t)*t^(a-1),t,z,inf)
> >
> >
> > lower Incomplete Gamma function (gammagreek in Maxima):
> >
> >  lower_gamma_incomplete(a,z)= integrate(exp(-t)*t^(a-1),t,0,z)
> >
> >
> Do we have a function to evaluate lower_gamma_incomplete?  Using
> gamma_incomplete is not very accurate if a is large and z is relatively
> small (about 1) since we end up subtracting two numbers approximately
> equal to gamma(a).

Thank you for the hint.

I have to look again at the numerical routines for the evaluation of the
Incomplete Gamma functions. We have two algorithm: a series expansion
and an expansion in continued fractions. Actual, the expansion in
continued fractions calculates the lower Incomplete Gammma function. The
last step in the evaluation is

gamma_incomplete(a,x) = gamma(a) - lower_gamma_incomplete(a,x)

to get the Incomplete Gamma function.

Dieter Kaiser

```