About computing integral integrate(exp(-x^k),x,0,1)

Hi everybody,

Thanks for checking my math on this Rich. In looking back at what I wrote I
noticed that I had omitted a factor of (1/k) in the transform of the
differential. The transform should look like dx = (1/k)*z^(1/k - 1)*dz. I'm
sorry if this might have caused confusion; I should monitor my typing more
carefully!

The version of the incomplete gamma function that I used is from Abramowitz
& Stegun (6.5.3); this is the version that I was familiar with from other
books. Using this definition I still get the answer that I originally got.
In the Maxima reference on this function the integral form of this function
follows A&S (6.5.3) but it is identified as A&S (6.5.2), an alternative
version. The situation is confusing as to what form Maxima really uses. I
think that your solution Rich corresponds to the integral form of
incomplete gamma from zero to x, vs my form from x to infinity. In doing a
numerical test on Maxima's gamma_incomplete(5,0) I get 24 = 4!, which seems
to correspond to the integral form that I used. So it seems to me that the
formula reference in Maxima's manual has a typo that is highly misleading.
I again apologize to the readers for any confusion that might have arisen
from any of my statements and extend my thanks to those, including you, who
attempted to clarify the problem.

Jim

On Wed, Jan 4, 2012 at 9:45 PM, Richard Hennessy
<rich.hennessy at verizon.net>wrote:

> I meant the integral becomes (1/k)*[gamma(1/k)-incomplete_**gamma(1/k,
> z)]. Typo.
>
> Rich
>
> -----Original Message----- From: Richard Hennessy
> Sent: Thursday, January 05, 2012 12:43 AM
> To: James Nesta ; maxima at math.utexas.edu
>
> Subject: Re: [Maxima] About computing integral integrate(exp(-x^k),x,0,1)
>
> "integral becomes(1/k)*[gamma(1/k)-**incomplete_gamma(1/k, 1)]"
>
> I think you made a mistake, it should be
>
> the integral becomes (1/k)*[gamma(1/k)-incomplete_**gamma(1/k, k)]
>
> Rich
>
> -----Original Message----- From: James Nesta
> Sent: Wednesday, January 04, 2012 10:48 PM
> To: maxima at math.utexas.edu
> Subject: Re: [Maxima] About computing integral integrate(exp(-x^k),x,0,1)
>
> In fooling with this integral I was able to get what I think is a
> manageable
> solution. As follows:
>
> Transform the variable x to z by the transformation x = z^(1/k), dx =
> z^(1/k-1)dz,
>
> the integral becomes(1/k)*[gamma(1/k)-**incomplete_gamma(1/k, 1)]. I think
> that Maxima should be able to handle each of these functions. I hope that
> this helps.
>
> Jim
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