1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

1>18 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

• EXAMPLE 12 .2 The function U (t) = It! , forte (-7r, 7r), extended periodi-
  cally by the equation U (t + 27r) = U (t), has the Fourier series representation

7r 4

00

1.

U(t)=ltl=-

2

• -2:. 2 cos((23- l)t].
  7r i = l (23 - 1)

Solution The function U (t) is an even function; hence we can use Theorem

12.3 to conclude that bn = 0 for all n and that

a; = -21"t COSJ "tdt = (2tsinjt. + --2cosjt)I" .2-

7r 0 1rJ 1rJ 0

_ 2cosj7r - 2 _ 2(-l)j - 2

• 7rj2 - 7rj2 for j = 1,2, ....

We compute the coefficient ao by

ao = -21" t dt = -t2 I" = 7r.

7ro 7ro

Using the ai and Theorem 12.3 produces the required solution.

12.1.1 Proof of Euler's Formulas

The following intuitive proof justifies the Euler formulas given in Equations

(1 2 -2) and (12-3). To determine ao we integrate both U (t) and the Fourier

series representation in Equation (12-1) from -7r to 7r, which results in

;: U (t) dt = ;: [ ~o + ~ (ai cosjt + b; sinjt)] dt.

Next, we integrate term by term to obtain

1 " 1"

00

1"

00

_" U (t) dt = a; _,,. 1 dt +{;a; _,. cosjt dt + {;b; 1" _,. sinjt dt.