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)|$$^{infinity}_{0}tex]+(1/2) (original integral except t^-1/2)<br /> <br /> after further integration, isn't it an endless cycle?$$

 

[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?