1550251515-Classical_Complex_Analysis__Gonzalez_

736 Chapter^9

while on y = ±m,

1 1 1

I csc 7l"ZI = < < - < 1

Vsin^2 7rX + sinh^2 7rm sinh7rm - 7rm

so that lf(z)I = 171" csc 7rzl < 7r for all z E C~. Hence, using (9.12-2), we

obtain

+oo

Z csc7r( = L (-It (2 ~n2

n=-cx:>

or changing ( into z,

1

00

2z

7l"CSC7l"Z = - + z I:C-lt z (^2) -n 2
n=l
1
00
=-+I:c-1r (1 -+-1)
z n=l z + n z - n
(9.12-4)

which converges absolutely for every z =/:-0, ±1, ±2, ... , and uniformly on

every compact set K that contains none of the integers.

3. Let f(z) = 7r cot 1l"Z and g(z) = l/z^2 n(n 2:: 1). Replacing z by 7l"Z

in (8.5-6), we have

for lzl < 1. Hence

Res 1l"Z cot 1l"Z • z-<^2 n+i) = (-It(27r)2n B^2 n

z=O (2n)!

and (9.12-2) gives

(9.12-5)

where the prime is used to indicate that the value v = 0 is to be omitted

from the summation. Since symmetric terms in this summation are equal,

we may write (9.12-5) in the form

1 (' ) 2 n (-1r-

1

B2n ~ 1 1 1

2 271" (2n)! = .i...J v2n = 1 + 22n + 32n + ...

v=l

(9.12-6)

which gives an expression of the Bernoulli numbers in terms of a p-series.

Conversely, (9.12-6) yields the evaluation of a p-series, with p even, if the