Complex Analysis - Maths KU

–1
y
Figure 4.7 Ln(z + 1) is not differentiable
on the ray shown in color.
x
4.1 Exponential and Logarithmic Functions 189
Solution (a) From the differentiation rulesof Section 3.1 we have that the
function zLn z isdifferentiable at all pointswhere both of the functionsz and
Ln z are differentiable. Because z isentire and Ln z isdifferentiable on the
domain given in (18), it followsthat zLn z isdifferentiable on the domain
defined by |z| > 0, −π