272 SPECIAL NUMBERS
trigonometric functions in terms of their hyperbolic cousins by using the rules sin z = -isinh iz ,
cos z = cash iz;
(6.86)
the corresponding power series are 2’ 23 25 sin2 = 1!-3!+5!--...
cosz
,
2’
23
25
sinhz = T+“j-i.+5r+...; .ci .; zi coshz = ol+2r+T+... . . .
20 22 24 = o!-2!+4?--...)
.
Hence cot z = cos z/sin z = i cash iz/ sinh iz = i coth iz, and we have
I see, we get “real”
functions by using imaginary numbers. (6.87)
Another remarkable formula for zcot z was found by Euler (exercise 73): zcotz = l-2tTg. k>,krr -z2
(6.88)
We can expand Euler’s formula in powers of z2, obtaining
.
Equating coefficients of zZn with those in our other formula, (6.87), gives us an almost miraculous closed form for infinitely many infinite sums: <(In) = H($) = (-l)np'
22n-1 n2nf3
(2n)! For
2n
’
integer n > 0.
(6.89)
example, c(2) = HE) = 1 + ; + ; +. . . = n2B2 = x2/6;
(6.90)
((4) = Hk) = 1 + & + & +. . . = -ff B4/3 = d/90.
(6.91)
Formula (6.89) is not only a closed form for HE), it also tells us the approximate size of Bzn, since H,,(ln) is very near 1 when n is large. And it tells US that (-l)n-l B2,, > 0 for all n > 0; thus the nonzero Bernoulli numbers alternate in sign.
