Approximating Gamma Function: Numerically Calculate \(\frac{3}{2}\)

[twalker40]

1. Numerically approximate \Gamma(\frac{3}{2}). Is it reasonable to define these as (\frac{1}{2})!?

2. Show in the sense of question 1. that (\frac{1}{2})! = \frac{1}{2}\sqrt{\pi} at least numerically.


How am i supposed to attempt this numerically? given that i do not know additional identities of the gamma function...

 

[Avodyne]

Hmm, seems like a poorly phrased question. The usual definition of the gamma function is

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

for \mathop{\rm Re}z>0. Then it's pretty easy to show that \Gamma(n)=(n-1)! when n is a positive integer. You could use this integral as a basis for numerical approximation, I suppose.

 

[Dick]

The definition of the gamma function is given by an integral. What is it? You can numerically approximate an integral. I think that's what they are after.

 

[twalker40]

The definition of the gamma function is given by an integral. What is it? You can numerically approximate an integral. I think that's what they are after.

okay, if i numerically approximate by plugging in 3/2 into the gamma function, i get infinity.

how am i supposed to use that information to arrive at the conclusion in #2?

 

[Dick]

No, you don't get infinity. Tell us how you did.

 

[twalker40]

O, I am sorry. I did my math incorrectly...

after reworking the problem, using integration by parts,
i'm stuck at
(-t^(1/2))/(e^t)|^{infinity}_{0}tex]+(1/2) (original integral except t^-1/2)<br /> <br /> after further integration, isn&#039;t it an endless cycle?

 

[Dick]

No, you aren't going to get much of anywhere integrating by parts. I thought you wanted to make a numerical approximation. Don't you just want to approximate the integral of t^(1/2)*exp(-t) from 0 to infinity?

 

[twalker40]

would that be a basic fnInt(y,x,0,99) command on the calc?

 

[HallsofIvy]

What function you use on a calculator depends on the calculator!

Are you required to do this using a specific calculator?