1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

cb-~Rter 6

integration

Overview

Of the two main topics studied in calculus-differentiation and integration-

we have so fur only studied derivatives of complex functions. We now turn

to the problem of integrating complex functions. The theory you will learn is

elegant, powerful, and a useful tool for physicists and engineers. It also connects

widely with other branches of mathematics. For example, even though the ideas

presented here belong to the general area of mathematics known as analysis, you

will see as an application of them one of the simplest proofs of the fundamental

theorem of algebra.

6.1 Complex Integrals

We introduce the integral of a complex function by defining the integral of a

complex-valued function of a real variable.

I Definition 6.1: Integral off (t)

Let f (t) = u (t) + iv (t), where u and v are real-valued functions of the real

variable t for a ~ t ~ b. Then

1bf(t)dt= 1bu(t)dt+i1bv(t)dt. (6-1)

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