Integral of Maxwell-Boltzmann dist.

[Denver Dang]

Homework Statement

I was wondering if there is some sort of trick to calculate the following:
$$\int{-{{e}^{-\varepsilon /{{k}_{B}}T}}}{{\varepsilon }^{3/2}}d\varepsilon \,$$
It's the derivative of the Maxwell-Boltzmann distribution, excluding the constants, times an energy in the power of (3/2).

Homework Equations

The Attempt at a Solution

I've found the solution from 0 to infinity if the energy was powered in n (n being an integer), but I haven't been able to find anything when it's a half-integer.

So is there some sort of trick to do this, or...?


Thanks in advance.

 

[Dick]

Homework Statement

I was wondering if there is some sort of trick to calculate the following:
$$\int{-{{e}^{-\varepsilon /{{k}_{B}}T}}}{{\varepsilon }^{3/2}}d\varepsilon \,$$
It's the derivative of the Maxwell-Boltzmann distribution, excluding the constants, times an energy in the power of (3/2).

Homework Equations

The Attempt at a Solution

I've found the solution from 0 to infinity if the energy was powered in n (n being an integer), but I haven't been able to find anything when it's a half-integer.

So is there some sort of trick to do this, or...?


Thanks in advance.

Since integer powers are nicer why not substitute $\epsilon=u^2$? Now you've got a gaussian times an integer power of u. That's a pretty well known problem.

 

[Denver Dang]

Ahhh yes, that should work. Don't know why I didn't think of that :/

Thank you :)