1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 737

corresponding Bernoulli number is known. For instance, since B 2 = 1/ 6

we have

rr^2 1 1 1

6 = 1 + 22 + 32 + 42 + ... (9.12-7)

Similarly, since B4 = -% 0 we find that

rr^4 1 1 1

90 =^1 + 2 4 + 3 4 + 4 4 + ... (9.12-8)

The reader will note that (9.12-6) implies that (-l)n-l B 2 n > 0: i.e., the

Bernoulli numbers B 2 n ( n ;:::: 1) are alternatively positive and negative.

4. Let f(z) = rrcscrrz and g(z) = 1/z^2 n. Replacing z by rrz in (8.5-8),

we have
00
(2^2 n - 2)B2
rrzcscrrz = 1 + L(-1t-^1 n rr^2 nz^2 n
n=l (2n)!

for lzl < 1. From this expansion it follows that

Resrrzcscrrz · z-(^2 n+l) = (-lt-^1 (2^2 n - 2)rr^2 n ~ B

z=O (2n)!

and since /3k = Resz=k rr csc rrz = (-l)k we have, by (9.12-2),

+co ( )k +oo ( )k+l

(-l)n-1(^2 2n _ 2)rr2n (2n B2n )! = _ '"'' L.J ~ k2n = '"'' L.J _-_1_ k2n

k=-oo k=-oo

which can be written as

~(22n2)rr2n (-l)n-lB2n = ~ (-l)k+l = 1-!__+_1 __ ··· (9.12-9)

2 (2n)! ~ k2n 22n. 32n

This equation gives another representation of the Bernoulli numbers, this
time in terms of an alternating p-series. Or we may use (9.12-9) to evalu-
ate the p-series (p even) if the corresponding Bernoulli number is already ,
known. For instance, for n = 1 we find that.

rr^2 1 1 1

12 =^1 - 2 2 + 3 2 - 4 2 + ... (9.12-10)

and for n = 2,

7rr^4 1 · 1 1

-=1--+---+···

720 24 34 44

(9.12-11)