- #1

- 3,003

- 6

## Homework Statement

I need to prove the relation:

[tex]

\Gamma(x+1) = e^{-x}x^{x+1/2}\int_{-\sqrt{x}}^\infty e^{-\sqrt{x}t}\left (1 +\frac{t}{\sqrt{x}}\right )^x\,dt \qquad(1)

[/tex]

## Homework Equations

Definition of Gamma function ?

[tex]

\Gamma(x) = \int_0^\infty t^{x - 1}e^{-t}\,dt

[/tex]

## The Attempt at a Solution

I have NO idea how to start this. I figured I should start out with the definition:

[tex]

\Gamma(x+1)=\int_0^\infty t^{x}e^{-t}\,dt \qquad(2)

[/tex]

But this does not do much. I tried letting u

^{2}= t -->dt = 2u*du and putting that back into (2):

[tex]

\Gamma(x+1) = 2\int_0^\infty u^{2x}e^{-u^2}u\,du \qquad(3)

[/tex]

Am I heading anywhere good?