       RE: Transformation of Gamma Function

• To: mathgroup at smc.vnet.net
• Subject: [mg31293] RE: [mg31266] Transformation of Gamma Function
• From: "Higinio Ramos" <higra at usal.es>
• Date: Sat, 27 Oct 2001 01:08:02 -0400 (EDT)
• References: <200110260828.EAA05608@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```----- Original Message -----
From: Andre Kronimus <kronimus at whu.edu>
To: mathgroup at smc.vnet.net
Subject: [mg31293] [mg31266] Transformation of Gamma Function

> Hi,
>
> when evaluating the integral (Exp[-Exp[-k t]-t], {t,0,infinity},{k>0})
> in Mathematica, it returns a transformed version including Gamma
> functions, namely Gamma[1+1/k]-Gamma[1/k,1]/k. The two expressions are
> identical for some numerical values that I checked. However, I have been
> unable to figure out which transformation rules Mthematica used for
> transforming the original integral. Does anybody have an idea which
> rules have been used or how to force Mathematica to output every
> intermediate step of the transformation?
>
> Andre
>
Doing the transformation E^(k t)=1/z, the original integral becomes in
In:=
Integrate[(1/k)Exp[-z]z^(1/k - 1) , {z, 0, 1}, Assumptions -> k > 0], whose
solution is:
Out=
\!\(\(Gamma[1\/k] - Gamma[1\/k, 1]\)\/k\),
and using the identity: Gamma[a+1]=a*Gamma[a],
results in the solution that Mathematica returns.
Higinio

```

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