Complex Analysis - Maths KU

4.1 Exponential and Logarithmic Functions 177
Theorem 4.1 Analyticity of e z
The exponential function e z isentire and itsderivative isgiven by:
d
dz ez = e z . (3)
Proof In order to establish that e z isentire, we use the criterion for analyticity
given in Theorem 3.5. We first note that the real and imaginary parts,
u(x, y) =e x cos y and v(x, y) =e x sin y, ofe z are continuousreal functions
and have continuousfirst-order partial derivativesfor all (x, y). In addition,
the Cauchy-Riemann equationsin u and v are easily verified:
∂u
∂x = ex cos y = ∂v
∂y
and
∂u
∂y = −ex sin y = − ∂v
∂x .
Therefore, the exponential function e z isentire by Theorem 3.5. By (9) of
Section 3.2, the derivative of an analytic function f isgiven by f ′ (z) = ∂u
∂x +
i ∂v
∂x , and so the derivative of ez is:
d
dz ez = ∂u ∂v
+ i
∂x ∂x = ex cos y + ie x sin y = e z . ✎
Using the fact that the real and imaginary parts of an analytic function
are harmonic conjugates, we can also show that the only entire function f that
agreeswith the real exponential function e x for real input and that satisfies
the differential equation f ′ (z) =f(z) isthe complex exponential function e z
defined by (1). See Problem 50 in Exercises 4.1.
EXAMPLE 1 Derivatives of Exponential Functions
Find the derivative of each of the following functions:
(a) iz 4 � z 2 − e z� and (b) e z2 −(1+i)z+3 .
Solution (a) Using (3) and the product rule (4) in Section 3.1:
d � 4
iz
dz
� z 2 − e z�� = iz 4 (2z − e z )+4iz 3 � z 2 − e z�
=6iz 5 − iz 4 e z − 4iz 3 e z .
(b) Using (3) and the chain rule (6) in Section 3.1:
d
�
e
dz
z2−(1+i)z+3 �
= e z2−(1+i)z+3 · (2z − 1 − i) .