# Integral definition of gamma function problem

• Sep 18th 2011, 11:17 PM
Potato
Integral definition of gamma function problem
Hey guys got a tricky question (for me at least) that i'm not sure about. Calculus was never really my strong point.

Not sure how to use the math tags very well either so i just attached a picture of my exam revision sheet. I get the basic gist of integration by parts and Laplace Transform functions but I can't seem to get this one going......

I seem to get Gamma Function Symbol (Can't remember the name) equals zero, now i know that's not right haha.

Some help would be great if you guys could manage.

Cheers

EDIT: Attached file : )
• Sep 18th 2011, 11:51 PM
CaptainBlack
Re: Integral definition of the gamma function problem
Quote:

Originally Posted by Potato
Hey guys got a tricky question (for me at least) that i'm not sure about. Calculus was never really my strong point.

Not sure how to use the math tags very well either so i just attached a picture of my exam revision sheet. I get the basic gist of integration by parts and Laplace Transform functions but I can't seem to get this one going......

I seem to get Gamma Function Symbol (Can't remember the name) equals zero, now i know that's not right haha.

Some help would be great if you guys could manage.

Cheers

EDIT: Attached file : )

Why is the subject line "Laplace transform", thit is not an LT question.

For (a) try integration by parts:

$\displaystyle \Gamma(x)=\int_0^{\infty} t^{x-1}e^{-x}\; dt= \int_0^{\infty} [t] \left[t^{x-2}e^{-x}\right]\; dt$

If you have any problems with that post a follow up question in this thread.

CB
• Sep 21st 2011, 12:28 AM
Potato
Re: Integral definition of gamma function problem

Thank you very much, a simple solution, I should have seen that straight away, funny how that works.

And I do apologise for the inaccurate title, I originally recognised the function as a Laplace Transform which is again, my mistake.

Thanks
• Nov 14th 2011, 10:44 PM
prime
Re: Integral definition of gamma function problem
can you please put the solution steps
• Nov 15th 2011, 01:03 AM
CaptainBlack
Re: Integral definition of gamma function problem
Quote:

Originally Posted by prime
can you please put the solution steps

What problems are you having with the integration by parts suggested up thread?

For the second problem, you know that $\displaystyle \Gamma (x)=(x-1)\Gamma(x-1)$ that means $\displaystyle \Gamma(x-1)=(x-2)\Gamma(x-2)$ and so on, therefore you get:
$\displaystyle \Gamma(x)=(x-1)\Gamma(x-1)=(x-1)(x-2)\Gamma(x-2)=(x-1)(x-2)(x-3)\Gamma(x-3)$
$\displaystyle \Gamma(x)=(x-1)(x-2)(x-3)...3.2.\Gamma(1)$
The only thing you have to do now is to prove that $\displaystyle \Gamma(1)=1$