# [Maxima] Integrating gamma_incomplete

Richard Hennessy rich.hennessy at verizon.net
Sat Mar 6 20:41:53 CST 2010

```This may be better, but is not tested fully.

matchdeclare([a,s],freeof(x),x,true,n,nonnegintegerp)\$
block([simp:false],tellsimp('integrate(gamma_incomplete(s, a*x^n), x),x*gamma_incomplete(s, a*x^n) -
(x*gamma_incomplete(1/n + s, a*x^n))/(a*x^n)^(1/n)))\$
remove([a,s,x,n],matchdeclare)\$

--------------------------------------------------
From: "Richard Hennessy" <rich.hennessy at verizon.net>
Sent: Saturday, March 06, 2010 9:01 PM
To: "Dieter Kaiser" <drdieterkaiser at web.de>
Cc: "Maxima List" <maxima at math.utexas.edu>
Subject: Re: [Maxima] Integrating gamma_incomplete

> This is not too hard.  I came up with this tellsimp rule that allows integration of gamma_incomplete of powers of x.
> I think it is correct.  You have to ratsimp() the result though.
>
> In pw.mac.
>
> matchdeclare(a,true,x,true,n,nonnegintegerp)\$
> block([simp:false],tellsimp('integrate(gamma_incomplete(s, x^n), x),x*(gamma_incomplete(s, x^n)) -
> (x*(gamma_incomplete(1/n + s, x^n)))/(x^n)^(1/n)))\$
> remove([a,x,n],matchdeclare)\$
> /* end of pw */
>
> (%i1) display2d:false;
> (out1) false
> (%i2) integrate(gamma_incomplete(s,x^15),x);
> (out2) gamma_incomplete(s,x^15)*x-gamma_incomplete(s+1/15,x^15)
> (%i3) diff(%,x);
> (out3) 15*x^(15*(s-14/15)+14)*%e^-x^15-15*x^(15*(s-1)+15)*%e^-x^15+gamma_incomplete(s,x^15)
> (%i4) ratsimp(%);
> (out4) gamma_incomplete(s,x^15)
> (%i5) integrate(gamma_incomplete(s,x^5),x);
> (out5) gamma_incomplete(s,x^5)*x-gamma_incomplete(s+1/5,x^5)
> (%i6) diff(%,x);
> (out6) 5*x^(5*(s-4/5)+4)*%e^-x^5-5*x^(5*(s-1)+5)*%e^-x^5+gamma_incomplete(s,x^5)
> (%i7) ratsimp(%);
> (out7) gamma_incomplete(s,x^5)
>
> HTH,
>
> Rich
>
>
> --------------------------------------------------
> From: "Richard Hennessy" <rich.hennessy at verizon.net>
> Sent: Saturday, March 06, 2010 2:10 PM
> To: "Dieter Kaiser" <drdieterkaiser at web.de>
> Cc: "Maxima List" <maxima at math.utexas.edu>
> Subject: Re: [Maxima] Integrating gamma_incomplete
>
>> I also noticed that maxima cannot integrate gamma_incomplete(s, +-x^n) for n=1,2,3,4, etc. which Mathematica can do.
>> I hope that can be done too.
>>
>> Rich
>>
>>
>> --------------------------------------------------
>> From: "Dieter Kaiser" <drdieterkaiser at web.de>
>> Sent: Saturday, March 06, 2010 7:49 AM
>> To: "Richard Hennessy" <rich.hennessy at verizon.net>
>> Cc: "Maxima List" <maxima at math.utexas.edu>
>> Subject: Re: [Maxima] Integrating gamma_incomplete
>>
>>> Am Freitag, den 05.03.2010, 22:07 -0500 schrieb Richard Hennessy:
>>>> I noticed Maxima can only integrate gamma_incomplete a couple times.
>>>> Mathematica can do it as many times as you want.  Is this a weakness
>>>> in integrate()?
>>>
>>> Maxima can only integrate the direct function gamma_incomplete, but not
>>> the case when a power is involved. Therefore, we get:
>>>
>>> (%i2) integrate(gamma_incomplete(a,x),x);
>>> (%o2) gamma_incomplete(a,x)*x-gamma_incomplete(a+1,x)
>>>
>>> We get a noun form, when we repeat the integration:
>>>
>>> (%i3) integrate(%,x);
>>> (%o3) 'integrate(gamma_incomplete(a,x)*x,x)
>>>       -gamma_incomplete(a+1,x)*x+gamma_incomplete(a+2,x)
>>>
>>> I have already proposed an extension on the mailing list
>>> integrals of the type x^v*gamma_incomplete(a,x). With this extension we
>>> will get:
>>>
>>> (%i5) integrate(gamma_incomplete(a,x),x);
>>> (%o5) gamma_incomplete(a,x)*x-gamma_incomplete(a+1,x)
>>>
>>> (%i6) integrate(%,x);
>>> (%o6) (gamma_incomplete(a,x)*x^2-gamma_incomplete(a+2,x))/2
>>>       -gamma_incomplete(a+1,x)*x+gamma_incomplete(a+2,x)
>>>
>>> (%i7) integrate(%,x);
>>> (%o7) ((gamma_incomplete(a,x)*x^3-gamma_incomplete(a+3,x))/3
>>>       -gamma_incomplete(a+2,x)*x+gamma_incomplete(a+3,x))
>>>       /2
>>>       -(gamma_incomplete(a+1,x)*x^2-gamma_incomplete(a+3,x))/2
>>>       +gamma_incomplete(a+2,x)*x-gamma_incomplete(a+3,x)
>>>
>>> All integrals are solved by Maxima.
>>>
>>> Dieter Kaiser
>>>
>>>
>>>
>>
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>
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```