534 Chaptersby making use of Corollary 8.1.- To expand f(z) = (1 + z)P in powers of z, where pis a given complex
number (in particular, a real number), and the notation (1 +z)P stands for
its principal value, namely, exppLog(l + z). This function is analytic in
D = C --{x: -oo < x:::; -1}, except when pis a nonnegative integer, in
which case f is analytic in C, or when pis a negative integer, in which case
f is a rational function with a pole at -1, and so analytic in C-{-1}. Since
jCn)(z) = p(p - 1) · · · (p - n + 1)(1 + z)P-n
we get
f(n)(O) = p(p - 1) · · · (p - n + 1) = n! (~)
by introducing the notation
(
p) = p(p - 1) · .. (p -n + 1)
n n!for n = 1, 2, ...
Thus noticing that f(O) = 1, we obtain the expansion
(8.3-12)called the binomial series. The series reduces to a polynomial when pis a
nonnegative integer since then (!) = 0 whenever n > p. Except for this
elementary case, we have R = 1, i.e., the resulting infinite series converges
on the unit disk lzl < 1.
- To expand f(z) = Arcsinz in power of z. Since f'(z) = (1-z^2 )-^112
(principal value), we see that the given function is analytic in C-{-1, 1}.
By using the binomial series to expand f'(z), we get
(1 -· z2)-l/2 = 1+f(-1li)c-z2r=1 + ~ z2 + 1. 3 z4 + ...
n=l n 2 2 · 4valid for lzl < 1. Now integrating along a rectilinear path from 0 to z
(lzl < 1) we have, recalling that ArcsinO = O,
co (-1/ ) z2n+1
Arcsinz = z + l:C-l)n^2 --n=l n 2n + 1
valid for lzl < 1.
1 z^3 1·3 z^5
=z+23+2.45+"· (8.3-13)