Obtain the Fermi function by comparing with the Bose-Einstein function

[Dixanadu]

Homework Statement

Hey guys,

So here's what we have:

Bose-Einstein function
$g_{v}(z)=\frac{1}{\Gamma(z)}\int_{0}^{\infty}\frac{x^{v-1}dx}{z^{-1}e^{x}-1}$

Fermi function
$f_{v}(z)=\frac{1}{\Gamma(z)}\int_{0}^{\infty}\frac{x^{v-1}dx}{z^{-1}e^{x}+1}$

And we have the series version of the Bose-Einstein function:

$g_{v}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^v}$

So by comparing the definitions of f and g, i have to find a similar series expansion for f.

Homework Equations

Given in the question!

The Attempt at a Solution

No idea where to start..i need a hint!

 

[member 553083]

Hint: Note that the integrals for f and g are quite similar, with the only difference being the sign of the denominator. If you consider the sign of the denominator as a multiplicative factor, then you can use the same approach to solve for the series expansion of f as well.