Sequences, Series, and Special Functions 547B2 2 Ba 3
=1+ -z + -z + ...
2! 3!
(8.5-4)But g(z) = (z/2)cothz/2 is an even function, i.e., g(-z) = g(~). Hence it
follows that Ba = Bs = · · · = B2k+1 = 0.The numbers Bn [sometimes Bi and the (-l)n-l B 2 n] are called the
Bernoulli numbers. They appear for the first time in Jakob Bernoulli's Ars
conjectandi (1713). From (8.5-4), namely,
00- coth Z Z - = 1 + L -( B2n )I z 2n ,
(^2 2) n=l 2n.
JzJ < 271" (8.5-5)
a number of other trigonometric and hyperbolic expansions can be derived.
For instance, replacing z by 2iz in (8.5-5), we find
oo 22n B
zcotz = 1 + f;C-lt ( 2 n)~n z
2n, Jzl < 71" (8.5-6)
andoo 22n(22n - l)B
tanz=cotz-2cot2z= ~(-1r-^1 (
2
n)!
2n z^2 n-l (8.5-7)
forJzl < ~71". Also,
zz csc z = z cot z + z tan 2
= 1 + ~(-1r-1 (22n - 2)B2n z2n
L.J (2n)! 'Jzl < 71" · (8.5-8)
n=lBy integrating both sides of (8.5-7) along a linear path from 0 to z (Jzl <
~ '11"), we get
oo 22n(22n - l)B
Logcosz = f;C-lt (2n)(2n)! 2n z2n, (8.5-9)From (8.5-6) we have1 ( 1) ~( l)n 22n B2n 2n-1
-; z cot z - = ~ - ( 2 n)! z ,^0 < Jzl < 71" (8.5-10)and since fz Log si~z = ~(zcotz -1), 0 < Jzl < 71", by integrating both
sides of (8.5-10) along a linear path from zo (0 < Jzol < '11") to z, we obtainL sinz _ L sinzo _ ~(-lt 22
nB2n ( 2n _ 2n)
og z og z 0 - ~ (2n)(2n)! z Zo