Complex Analysis - Maths KU

– π
2
–3
– π
3
–2
y
2
1.5
1
0.5
–
π
6
–0.5
–1
–1
–1.5
–2
2
1
–1
–2
π
6
w = sin z
π
3
π
2
The mapping w = sin z. See page 208.
v
1 2 3
x
u
4
Elementary
Functions
4.1 Exponential and Logarithmic Functions
4.1.1 Complex Exponential Function
4.1.2 Complex Logarithmic Function
4.2 Complex Powers
4.3 Trigonometric and Hyperbolic Functions
4.3.1 Complex Trigonometric Functions
4.3.2 Complex Hyperbolic Functions
4.4 Inverse Trigonometric and Hyperbolic Functions
4.5 Applications
Chapter 4Review Quiz
Introduction In the last chapter we defined a
class of functions that is of the most interest in
complex analysis, the analytic functions. In this
chapter we shall define and study a number of
elementary complex analytic functions. In particular,
we will investigate the complex exponential,
logarithmic, power, trigonometric, hyperbolic,
inverse trigonometric, and inverse hyperbolic
functions. All of these functions will
be shown to be analytic in a suitable domain
and their derivatives will be found to agree with
their real counterparts. We will also examine
how these functions act as mappings of the complex
plane. The set of elementary functions will
be a useful source of examples that will be used
for the remainder of175 this text.