H.W. 3 (a) The Bernoulli numbers $B_0, B_1, B_2, \dots$ are defined by
$B_n = n!c_n$
where
$\begin{cases} \frac{z}{e^z - 1}, & z \neq 0 \\ 1, & z = 0 \end{cases} = \sum_{n=0}^{\infty} c_n z^n$.
Note that $f(z)$ is analytic at $z = 0$ since, for all $z$,
$\frac{z}{e^z - 1} = \frac{z}{z + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots} = \frac{1}{1 + \frac{z}{2!} + \frac{z^2}{3!} + \dots}$
Perform long division on the right-hand quotient to show that $B_0 = 1$, $B_1 = -1/2$, $B_2 = 1/6$.
(b) Show that the coefficients of odd order beyond 1, i.e., $B_3, B_5, B_7, \dots$ are all zero.
<Hint> $f(z) + z/2 = (z/2)\cosh(z/2)/\sinh(z/2)$ is an even function of z. See Problem 30,
Section 5.4, Textbook.
(c) Where is the series expansion of Part (a) valid?
