Question

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?

          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?

H.W. 3 (a) The Bernoulli numbers B0, B1, B2, … are defined by
Bn = n!cn
where
(z)/(e^z - 1), z ≠ 0
1, z = 0 = ∑n=0^∞ cn z^n.

Note that f(z) is analytic at z = 0 since, for all z,
(z)/(e^z - 1) = (z)/(z + (z^2)/(2!) + (z^3)/(3!) + …) = (1)/(1 + (z)/(2!) + (z^2)/(3!) + …)
Perform long division on the right-hand quotient to show that B0 = 1, B1 = -1/2, B2 = 1/6.
(b) Show that the coefficients of odd order beyond 1, i.e., B3, B5, B7, … 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?

Added by Tracy W.

Question

Please give Ace some feedback