1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

520 CHAPTER 1 2 • FOURIER SERIES AND THE LAPLACE TRANSFORM

Therefore, we can use the results of Equations (12-6)-(12-9) in Equation (12-5)
to obtain

J~ U (t) cos mt dt = 7ram, for m = 0, 1, .. .,

establishing Equation (12-2). We leave as an exercise for you to establish Euler's
second formula, Equation (12-3), for the coefficients {bn}· A complete discussion
of the details of the proof of Theorem 12.1 is available in some advanced texts.
See, for instance, John W. Dettman, Applied Complex Variables, Chapter 8,
Macmillan, New York, 1 965.

-------~EXERCISES FOR SECTION 12.1

For Exercises 1- 2 and 6- 11, find the Fourier series representation.

1. U (t) = { 1, for 0 < t < 11';
   
   
   • 1, for - 7r<t<O.
     The graph of U ( t) is shown in Figure 12.4.
   
   
   
   

• n

s

f s = U(t)

I

2 "

Figure 12 .4

1t

2. v (t) = { j -t,

2 +t,

for 0 ::; t ::; ir;

for - 11' < t < O.

The graph of V ( t) is shown in Figure 12.5.

s

Figure 12 .5