Functions that integrate to a gamma function?

Main Question or Discussion Point

maple syntax:

int(theta^y * exp(-theta*(1-alpha) ) , theta)

I have a distribution that I need to integrate, and I know the result should have a gamma function in it.

The only thing I have found helpful is:
http://en.wikipedia.org/wiki/Gamma_function

My function is kind of in that form (theta^something * exp(something) ), but the "somethings" dont seem to be able to be manipulated into that form. Any hints on how to go about this?

Thanks!

David

You could match it with this generalized form of the gamma function (sometimes called the Plica Function):

$$\Gamma (a,z)=\int_z^{\infty } t^{a-1} e^{-t} dt$$

Or you could also match it with the exponential integral function (n is an integer):

$$E_n (z)=\int_1^{\infty } \frac{e^{-z t}}{t^n} dt$$

If you must match it to a non-generalized gamma expression, show us the final form you are aiming for.

I dont understand how I would match it with those? The problem is I have

t^alpha

and I need
t^(alpha - 1)

The only way I would know to "mold" mine into the correct one is to multiply by t^(-1) and t, then take t outside the integral and combine t^alpha with t^(-1) to get t^(alpha-1). However, t is the integration variable, so I can't do that!!

any other thoughts?

Thanks,

David

Mute
Homework Helper
Use the recursive property of the gamma function:

$$\Gamma(\alpha+1) = \alpha \Gamma(\alpha)$$

This does require, however, that your limits of integration are 0 to infinity.

I dont understand how I would match it with those? The problem is I have

t^alpha

and I need
t^(alpha - 1)

However, t is the integration variable
Use integration by parts, differentiate t^alpha.

If the integral you are looking for is the following (assuming the limits are correct):

$$\int_{0}^{\infty}\theta^y \cdot e^{-\theta(1-\alpha)} d\theta$$

You can transform it using:

$$\theta(1-\alpha)=u$$

This gives you something related to the Gamma function.