1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.l • FOURIER SERIES 521

3. For Exercises 1 and 2, verify that U (t) = -V' (t) by termwise differentiation of
   the Fourier series representation for V (t).

00 ( li-1

4. For Exercise 1, set t = ~ and conclude that ~ = 2::: ;. _
   j=l J^1

2 00 1

5. For Exercise 2, set t = 0 and conclude that ~ = 2:::. 2.
   j=l (23 -1)

{

-1

6. u (t) = 1, '

-1,

for! < t < 7r;

for- 2 n<t<~;

for - 7l' < t < - 2 n.

The graph of U (t) is shown in Figure 12 .6.

s

o--......... 1----<> s = U(t)

• 1t _!!;
  2

0----0 - 1

Figure 12. 6

{

7r - t

7. u (t) = t, '
   -7r - t,

TC

I

1t

for~<t:S7r,

for - 2 " < t :S! ,

for -7r :S t :S - 2 " •

The graph of U (t) is shown in Figure 12.7.

s

Figure 12.7