Complex Analysis - Maths KU

188 Chapter 4 Elementary Functions
z = 0 isa branch point. As the following theorem demonstrates, the branch
f1 isan analytic function on itsdomain.
Theorem 4.4 Analyticity of the Principal Branch of ln z
The principal branch f1 of the complex logarithm defined by (19) isan
analytic function and itsderivative isgiven by:
f ′ 1(z) = 1
. (20)
z
Proof We prove that f1 isanalytic by using the polar coordinate analogue to
Theorem 3.5 of Section 3.2. Because f1 isdefined on the domain given in (18),
if z isa point in thisdomain, then we can write z = re iθ with −π