1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

536 CHAPTER, 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

Hint: F(t)=:'.!:.

8

+~E.

1

2 cos[(2j-1)t]+~E

1
7r ;=I (2J -1) 7r ; : I 22 (2J - J)^2 cos(2(2j- l)t],
where a4n = 0 for all n.

12 .4 The Fourier Transform

If we let U (t) be a real-valued function with period 211", which is piecewise con-
tinuous such that U' (t) also exists and is piecewise continuous, then U (t) has
the complex Fourier series representation

n=-oo

where

Cn = - 1 j" u (t) e- intdt,

27r -tr

for all n.

The coefficients {en} are complex numbers. Previously, we expressed U (t) as
the real trigonometric series
00
U (t) = a
2

o + L (a,,. cosnt + bnsinnt).

n=l

(12-19)

Hence a relationship between the coefficients is

a,.= Cn + e-n, for n = 0, 1, ... , and

bn = i (Cn -e- n), forn=l,2, ....

We can easily establish t hese relations. We start by writing

00 00
U (t) = Co+ L Cneint + L c_ne-int (12-20)
n = l n=l
00 00

=Co+ L Cn(cosnt+isinnt)+ L:e-n(cosnt-isinnt)

n=l n = l

00

= Co+ L [(en+ e-n) cosnt + i (en - C-n) sin nt].

n=t

Comparing Equations (12-20) and (12-19), we see that ao = 2co, an= Cn + c_n,
and bn = i (en -C- n )·
If U (t) and U' (t) are piecewise continu?us and have period 2L, then U (t)
has the complex Fourier series representation
00
U(t) = L Cnei trnt/L, (12-21)
n=-oo