1550251515-Classical_Complex_Analysis__Gonzalez_

7

Integration

7.1 INTRODUCTION

In this chapter we consider the definition and properties of the integrals of
continuous complex functions along various types of paths of integration.
In particUlar, we discuss the case where the integrand is analytic in a certain
region. Also, we consider some results concerning the integration of some
simple types of discontinuous functions.

7 .2 INTEGRAL OF A COMPLEX FUNCTION OF A

REAL VARIABLE

Definition 7.1 A function F(z) such that F'(z) = f(z) in some set A
is called a primitive function of f(z), and also an antiderivative or an
indefinite integral of f(z) in A..

Definition 7 .2 The definite integral of a continuous complex function
f(t) = u(t) + iv(t) of the real variable t over an interval [a, b] is defined by

lb f(t) dt =lb u(t) dt + i lb v(t) dt (7.2-1)

Theorem 7 .1 The definite integral, as defined by (7 .2-1) has the following
properties:.

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