1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

c a ter 12

our1 r series anll

the I place

transform

Overview

In this chapter we show how Fourier series, the Fourier transform, and the
Laplace transform are related to the study of complex analysis. We develop
the Fourier series representation of a real-valued function U (t) of the real vari-
able t. We then discuss complex Fourier series and Fourier transforms. Finally,
we develop the Laplace transform and the complex variable technique for finding
its inverse. In this chapter we focus on applying these ideas to solving problems
involving real-valued functions, so many of the theorems throughout are stated
without proof.

12.1 Fourier Series

Let U ( t) be a real-valued function that is periodic with period 271"; that is,

U(t+21r) = U(t), for all t.

One such function is s = U (t) = sin (t -~) + 0.7 cos (2t - 7r - D + 1.7. Its

graph is obtained by repeating the portion of the graph in any interval of length

271", as shown in Figure 12.1.

s

• Zll It 211 311 411
  Figure 12.1 A function U with period 271".

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