Mathematics of Physics and Engineering

Laurent Series 215

is specified by the condition Lnl = 0. More generally, by fixing the value
of ln z at one point ZQ ^ 0, we define a branch of the natural logarithm.
For example, condition lnl = 2m results in a different branch / = f(z) of
ln z, such that f(i) = (5ir/2)i. The values of different branches at the same
point vary by an integer multiple of 27ri.

EXERCISE 4.3.15.C (a) Use the Cauchy-Riemann equations in polar coordi-
nates to verify that the derivative of Ln z (and, in fact, of every branch of
the natural logarithm) is 1/z. (b) Verify that —Ln(l — z) = Y^k>\ zk/k- (If
everything else fails, just integrate term-by-term the expansion of 1/(1 — z).)
Using the exponential and the natural logarithm, we define the complex
power of a complex number:

zw = ewlnz. (4.3.18)

While the natural logarithm Inz has infinitely many different values, the
power can have one, finitely many, or infinitely many values, depending on
w. Quite surprisingly, a complex power of a complex number can be real:

jt = ei(ni/2+2*ki) = e-^/2-27Tfc) k = Q) ±1) ±2)... f

which was first noticed by Euler in 1746. By selecting a branch of the nat-
ural logarithm, we select the corresponding branch of the complex power.
For example il — e-7r/^2 corresponds to the principal value of the natural
logarithm. What is the value of il if we take ln 1 = —2-KI>.

EXERCISE 4.3.16. (a)c Verify that if m is an integer, and w = 1/m,
then the above definition of zw is consistent with (4-1.5), page 185. (b)B
For what complex numbers w does the expression zw have finitely many
different values? Hint: recall that the exponential function is periodic with period
2m. (c)A Assume that in (4-3.18) we take the principal value of the natural
logarithm. Verify that, for every complex number w, the corresponding
function f(z) = zw is analytic at every point z/0, and f'(z) = wzw~^1 •
Convince yourself that this is true for every branch of the complex power.

4.4 Singularities of Complex Functions

4.4.1 Laurent Series

We know that a function analytic at a point ZQ can be written as a Taylor
series that converges to the values of the function in some neighborhood