Sequences, Series, and Special Functions 533If n = 2m the coefficients vanish, and if n = 2m + 1, the coefficients
reduce to
1 2i^2 m+l
----
(2m + 1)! 2i
Hence we get=(-1r
(2m + 1)!(^00) ( l)m 3 5
sin z = """" - z^2 m+l = z - ~ + ~ - ...
m=O L.J (2m + 1)!. 3! 5!(8.3-9)again with R = oo. That the two series can be combined as shown for
any value of z for which both converge follows from Theorem 4.12. See
also Theorem 8.6.
By term-by-term differentiation of the series in (8.3-9) we obtain
(Corollary 8.5, Example):~ (-1r 2 z2 z4
cosz = ~ ( 2 m)! Z m = 1- 21 + 41 - · · · (8.3-10)with R = oo.
- To expand f(z) = Log(l + z) in power of z. The given function is
analytic in the region D = C - {x: -oo < x::; -1}. Hence the nearest
singularity off to the origin is s = -1, so the desired expansion has radius
of convergence R = 1. We have
J'(z) = (1 + z)-^1
J"(z) = (-1)(1 + z)-^2f(n)(z) = (-1)(-2) · · · [-(n - 1)](1 + z)-n
Hence, f(O) = Log 1 = 0, f'(O) = 1, J1'(0) = -1!,
(-l)n-^1 (n - 1)! and we get
Log( 1 + z) = z - ~ z^2 + .. · + ( -1 r-^1 .!_ z n + .. ·
2 · n.valid for Jzl < 1. Otherwise, from
... '
f'(z)= _1_ =1-z+z2-···+(-1r-1zn-1+···
l+z(8.3-11)which converges in D = {z: lzl < 1}, and uniformly on any smaller disk,
we get by term-by-term integration along a rectilinear path connecting 0
to z E D,
1
z --dz = L ( og 1 + z ) L - og 1 = z - - (^1) z··? + · · · + -( l)n-1 - (^1) z n + · · ·
0 l+z^2 n