Complex Analysis - Maths KU

156 Chapter 3 Analytic Functions
EXAMPLE 4 A Function Differentiable on a Line
In Example 2 we saw that the complex function f(z) =2x 2 + y + i(y 2 − x)
was nowhere analytic, but yet the Cauchy-Riemann equations were satisfied
on the line y =2x. But since the functions u(x, y) =2x 2 + y, ∂u/∂x =4x,
∂u/∂y =1,v(x, y) =y 2 − x, ∂v/∂x = −1 and ∂v/∂y =2y are continuous at
every point, it follows that f is differentiable on the line y =2x. Moreover,
from (9) we see that the derivative of f at points on this line is given by
f ′ (z) =4x − i =2y − i.
The following theorem is a direct consequence of the Cauchy-Riemann
equations. Its proof is left as an exercise. See Problems 29 and 30 in Exercises
3.2.
Theorem 3.6 Constant Functions
Suppose the function f(z) =u(x, y)+iv(x, y) is analytic in a domain D.
(i) If|f (z)| is constant in D, then so is f(z).
(ii) Iff ′ (z) =0inD, then f(z) =c in D, where c is a constant.
Polar Coordinates In Section 2.1 we saw that a complex function
can be expressed in terms of polar coordinates. Indeed, the form f(z) =
u(r, θ) +iv(r, θ) is often more convenient to use. In polar coordinates the
Cauchy-Riemann equations become
∂u
∂r
1 ∂v
=
r ∂θ ,
∂v
∂r
= −1
r
∂u
. (10)
∂θ
The polar version of (9) at a point z whose polar coordinates are (r, θ) is
then
f ′ (z) =e −iθ
� �
∂u
+ i∂v =
∂r ∂r
1
r e−iθ
� �
∂v
− i∂u . (11)
∂θ ∂θ
See Problems 33 and 34 in Exercises 3.2.
Remarks Comparison with Real Analysis
In real calculus, one of the noteworthy properties of the exponential function
f(x) =e x is that f ′ (x) =e x . In (3) of Section 2.1 we gave the definition
of the complex exponential f(z) =e z . We are now in a position
to show that f(z) =e z is differentiable everywhere and that this complex
function shares the same derivative property as its real counterpart, that
is, f ′ (z) =f(z). See Problem 25 in Exercises 3.2.