Stirling's Approximation Problem

[Rockoz]

Homework Statement

Use Stirling's Approximation formula to evaluate the following:
1) Γ(12.3) where Γ(N) is the gamma function
2) (π^2)! This is suppose to be pi squared
3) (1/2)!

Homework Equations

So here is Stirling's approximation formula:

N! ~= sqrt(2πN)(N/e)^N (where ~= means approximately equal to). And this is accurate to at least two decimal places for N >= 9.

The Attempt at a Solution

For the first problem I am unsure about what to do. I found a formula that says Γ(n) = (n-1)! but I'm not sure if this is only valid for integers. Can I simply plug into stirling's approximation formula (n-1) or in this case (12.3 - 1) = 11.3 to obtain the approximation for Γ(12.3)? For the other two problems do I simply plug in pi^2 and (1/2) into the formula? I apologize if this seems overly simple but this my first time ever seeing this formula and want to make sure I know how to use it in general. Thank you very much to anyone who can help.

 

[Simon Bridge]

I found a formula that says Γ(n) = (n-1)! but I'm not sure if this is only valid for integers.

That version is only valid for positive integers.

The Gamma function is more broadly defined for complex numbers other than non-positive integers.

Once you have the Stirling approximation to the gamma function, you can use it to evaluate the rest... it's not just a matter of blindly plugging the number into the equation, you have to make the number the right kind to fit.