1550251515-Classical_Complex_Analysis__Gonzalez_

7 40 Chapter 9

9.13 The Logarithmic Derivative

Definition 9.13 The logarithmic derivative of J(z) is defined to be

d f'(z)

dz logf(z) = f(z) (9.13-1)

at all points where f is analytic and different from zero. In (9.13-1) it is

immaterial what particular branch of the logarithmic function is considered.

Theorem 9.14 The logarithmic derivative of the product of two analytic
functions is the sum of the logarithmic derivatives of the factors at all
points of a common domain of definition where neither factor vanishes.

Proof In fact, we have

d d f' 91

dz log(! g) = dz (log f + log g) = f + g

9.14 ZEROS AND POLES OF MEROMORPHIC

]~UNCTIONS

Theorem 9.15 The logarithmic derivative of a nonconstant meromor-

phic function f in a region R is a meromorphic function in R with simple

poles at the zeros and poles off, with residues +a or -a, a denoting the

multiplicity of the corresponding zero or pole.

Proof If a is a zero of order a off, we have

f(z) = (z -a)°'g(z)

where g( z) is analytic and different from zero at a. Hence in a deleted

neighborhood of a we have

f'(z) = ~ + g'(z)

f(z) z-a g(z)

Since g'(z)/g(z) is analytic at a, this shows that a is a simple pole of f'/f

with residue a. Similarly, if a is a pole of order a off, we have

f(z) = (z -a)-°'g(z)

where again g( z) is analytic and different from zero at a. It follows that

f'(z) = -a + g'(z)

f(z) z - a g(z)