1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12. l • FOURIER SERIES 515

Definition 12.2: Fourier series

HU (t) is periodic with period 211' and is piecewise continuous on [-11', 11'), then
the Fou r ier series S ( t) for U ( t) is

00

S (t) = a; + L (a; cosjt + b; sinjt),

j=l

(12-1)

where the coefficients a; and b; are given by Euler's formulas:

a; = -^1 j" U (t) cosjt dt,

11' -1'

for j = 0, 1, ... , ( 12 -2)

and

b; = .!. j" U (t) sinjt dt,

11' -1'

forj=l,2, .... (^12 -3)

We introduced the factor ~ in the constant term ~ on the right side of Equa-
tion ( 12 -1) for convenience so that we can obtain <lQ from the general formula
in E<iuation (12-2) by setting j = 0. We explain the reasons for this strategy
shortly. Theorem 12.1 deals with convergence of the Fourier series.

• EXAMPLE 12.1 The function U (t) = ~ fort E (-11', 11'), extended periodi-
  cally by the equation U (t + 211') = U (t), has the Fourier series expansion

00 (-1);+ 1

U (t) = L. sinjt.

j=l J