Complex Analysis - Maths KU

z 0
C
z k *
z 1 * z 2 * z 2
z 1
z k
z k–1
Figure 5.19 Partition of curve C into
n subarcs is induced by a partition P
of the parameter interval [a, b].
z n
5.2 Complex Integrals 247
integral. The following list of assumptions is a prelude to the definition of a
complex integral. For the sake of comparison, you are encouraged to review
the lists given prior to Definitions 5.1 and 5.2. Also, look over Figure 5.19 as
you read this new list.
Steps Leading to the Definition of the
Complex Integral
1. Let f be a function of a complex variable z defined at all points on a
smooth curve C that lies in some region of the complex plane. Let C
be defined by the parametrization z(t) =x(t)+iy(t), a ≤ t ≤ b.
2. Let P be a partition of the parameter interval [a, b] into n subintervals
[tk−1, tk] of length ∆tk = tk − tk−1:
a = t0