Functions that integrate to a gamma function?

  • Thread starter daviddoria
  • Start date
95
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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
 

Answers and Replies

1,222
2
You could match it with this generalized form of the gamma function (sometimes called the Plica Function):

[tex]\Gamma (a,z)=\int_z^{\infty } t^{a-1} e^{-t} dt[/tex]

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

[tex]E_n (z)=\int_1^{\infty } \frac{e^{-z t}}{t^n} dt[/tex]

If you must match it to a non-generalized gamma expression, show us the final form you are aiming for.
 
95
0
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
1,381
10
Use the recursive property of the gamma function:

[tex]\Gamma(\alpha+1) = \alpha \Gamma(\alpha)[/tex]

This does require, however, that your limits of integration are 0 to infinity.
 
1,222
2
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.
 
279
0
If the integral you are looking for is the following (assuming the limits are correct):

[tex]\int_{0}^{\infty}\theta^y \cdot e^{-\theta(1-\alpha)} d\theta[/tex]

You can transform it using:

[tex]\theta(1-\alpha)=u[/tex]

This gives you something related to the Gamma function.
 

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