1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

516 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

s

Figure 12.3 T he function U (t) =;,and the approximations S1 (t), S2 (t), and Ss (t).

Solution Using Equation ( 12-2} and integrating by parts, we obtain

11" t. (tsinjt cosjt) I"

a;=; ,. 2 COSJt dt = 2 7rj + 2 7rj2 ,, = 0 ,

and then using Equation (12-3) we get

b ;=-11" -SlllJtt= t.. d (-tcosjt + - -sinjt) I"

'Tr -7< 2 27rj 27rj2 -1f

• COSj'lr (- l)j+l
  j j
  We compute the coefficient a 0 by

1 1" t t

2

ao = - - dt = -!'.'.." = 0.

7r -1f 2 47r

Substituting the coefficients a; and b; into Equation (12- 1) produces the required
solution. The graphs of U (t) and the first three partial sums: 81 (t) = sin t,

S2 (t) = sint - ~sin 2t, and S3 (t) =sin t - ~sin 2t + ~sin 3t. These sums are

shown in Figure 12.3.

We now state some general properties of Fourier series that are useful for

calculating the coefficients. We leave the proofs for you.