Fall 2006 9 Math

Consider the function $F_k (\eta)$ defined by the integral(1)
\begin{align} F_{k} \left ( \eta \right ) = \int^{\infty}_{0} \frac{x^k}{e^{x-\eta}+1}, \:\: k=0,1,2,3,... \end{align}

Find an analytic (polynomial in $\eta$) expression for

(2)
\begin{align} F_2 \left ( \eta \right ) -F_2 \left ( - \eta \right ) \end{align}

Hint: By integrating by parts, show that

(3)
\begin{align} \frac{d F_k (\eta)}{d\eta} = kF_{k-1}(\eta) \end{align}

and start with F0($\eta$). You are given

(4)
\begin{align} F_1 (0) = \frac{\pi^2}{12} \end{align}