532 Chapters
The uniqueness of the expansion (8.3-5) follows from the identity prin-
ciple for power series (Theorem 4.23). This implies that the power series
expansion can be obtained equally well by other means. See Examples 3,
4, and 6.
Examples 1. To expand f(z) = (z - 2)/(z + 2) in powers of z + 1.
The given function is analytic in C except at the point z = -2, where it
has a pole. In this case a = -1, so the resulting expansion will have a radius
of convergence R = I - 2 + ll = 1. The successive derivatives of f(z) are:
Hence
J'(z) = 4(z + 2)-^2
J"(z) = 4(-2)(z + 2)-^3J<n>(z) = 4(-2)(-3) · · · (-n )( z + 2)-(n+l)
= (-1r-^1. 4n!(z + 2)-(n+i)f(-1) = -3, J'(-1) = 4, J"(-1) = (-1)4. 2!,
J(n)(-l) = (-l)n-14n!and by using (8.3-1), we get
... ,
z - ~ = -3 + 4(z + 1) - 4(z + 1)^2 + ... + (-1r-^1 4(z + 1r + · · ·
z+2valid for lz + ll < 1.
- To expand f(z) = ez in powers of z. Here J(n)(z) = ez and
J(n)(O) == 1. Hence by (8.3-7) we have
(8.3-8)Since ez is an entire function the series above converges for all finite values
of z; i.e., we have R = oo.
- To expand f(z) = sinz in powers of z. As before, we may obtain the
desired result by using the Cauchy-Maclaurin expansion (8.3-7). However,
in view of sinz = (eiz - e-iz)/2i and the expansion (8.3-8) we have
sinz = ~ [~ 2_ (izt -~ 2_ (-izt]
2i L.t n! L.t n!
n=O n=O