Asymptotic Expansion Exercise: Finding a Gaussian Point Solution

[MementoMori96]

Homework Statement

Hi, i have this exercise

and i have to find the asymptotic expansion on a gaussian point .

Homework Equations

The Attempt at a Solution

[/B]
I have transformed x^t in e^{t ln x} in order to have the form used in the formula but ln x is not regular in 0 and it give some problems and moreover it hasn't a maximum . How can i do ? Thanks

 

[Ray Vickson]

Homework Statement

Hi, i have this exercise

View attachment 213965
and i have to find the asymptotic expansion on a gaussian point .

Homework Equations

The Attempt at a Solution

[/B]
I have transformed x^t in e^{t ln x} in order to have the form used in the formula but ln x is not regular in 0 and it give some problems and moreover it hasn't a maximum . How can i do ? Thanks

By a simple change of variable you can express your function $f(t) = \int_0^{\infty} x^{t-1} e^{-\pi x} \, dx$ in terms of the standard Gamma function
$$ \Gamma(z) \equiv \int_0^{\infty} y^{z-1} e^{-y} \, dy$$
Then, you can find methods for asymptotic expansion of $\Gamma$ in hundreds of web pages (or good old-fashioned books). For example, the link http://mathworld.wolfram.com/GammaFunction.html
has everything you need.

 

[fresh_42]

I suggest to substitute $y=\pi x$ in order to sort out the constants in your equation. Then you are left with the gamma function. What do you know about it, resp. which approximations are you allowed to use? If none, then look up proofs on the internet for its asymptotic behavior.