Mathematics of Physics and Engineering

Exponential Function 213

of z. Differentiating term-by-term the power series for g gives f{z) =

Ek>i(-l)k+lk(z ~^1 )k~^1 = Efc>o(-^1 )fe(fc + 1)(* - !)fc. where in the last

equality we changed the summation index to start from zero. The series

converges for \z — 1| < 1.

EXERCISE 4.3.10. c Find the power series expansions of the following
functions at the given points, and find the radius of convergence: (a)
f(z) — z + z + l at ZQ = i. Hint: put z = i+(z — i) and simplify. Alternatively,
put w = z — i so that z = w + i, and find the expansion of the resulting function in
powers of w; then replace w with (z — i). (b) f(z) = l/(z + 1z + 2) at ZQ = 0
Hint: use partial fractions: z^2 + 2z + 2 = (z + l)^2 + 1 = (z + 1 — i)(z + 1 + i),
f(z) = A/(z + 1 - i) + B/(z + l + i).

Given an analytic function, you usually do not need the explicit form of

its Taylor series to find the radius of convergence of the series. Indeed,

by Theorem 4.3.4, the radius must be the distance from ZQ to the closest

point at which / is not analytic (think about it...). FOR EXAMPLE, the

function f(z) = l/(z^2 + 2z + 2) is not analytic only at points z = l±i, and

both points are y/2 away from the origin. As a result, the Maclaurin series

for / has the radius of convergence equal to \pl.

EXERCISE 4.3.11.c Consider the function f(z) = (. 2 w 2 _ 5 s at the point

ZQ = 1 + 2i. (a) Without computing the series expansion, find the radius of

convergence of the Taylor series of f at ZQ. (b) Find the series expansion

and verify that it has the same radius of convergence as you computed in

part (a).

4.3.3 The Exponential Function

For z e C, we define

e

z

= Zw^ = E

[

-Tdrw>

C0SZ =

^(kr-

(4

3

fc>0 fc>0 V ' fe>0 V '

EXERCISE 4.3.12. (a)c Verify that all three series in (4-3.13) converge

for every z € C. (b)B Verify the two main properties of the exponential

function:

ezi+z* =e'ie*2t (e«y==e*)