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Chapter 1 Fourier Series and Integrals 63

Figure 2 f(x)=x,−π<x<π,fperiodic with period 2π.

Thus, for this function, we have

f(x)∼

∑∞

n= 1

2 (− 1 )n+^1

n sin(nx)

∼ 2

(

sin(x)−^1

2

sin( 2 x)+^1

3

sin( 3 x)−···

)

.

The Appendix contains some integration formulas that are convenient for
finding Fourier coefficients. It is also useful to know these special values of
sines and cosines that come up frequently in Fourier series.

sin(nπ)= 0 , cos(nπ)=(− 1 )n, forn= 0 ,± 1 ,± 2 ,...,

sin

(

( 2 n− 1 )π

2

)

=(− 1 )n+^1 , cos

(

( 2 n− 1 )π

2

)

= 0 ,

forn= 0 ,± 1 ,± 2 ,....

Note that the second line involves onlyoddmultiples ofπ/2. Even multiples
ofπ/2 are included in the first line.

EXERCISES

1. Find the Fourier coefficients of the functions given in what follows. All are
   supposed to be periodic with period 2π. Sketch the graph of the function.
   a.f(x)=x, −π<x<π;
   b.f(x)=|x|, −π<x<π;
   c. f(x)=

{ 0 , −π<x<0,

1 , 0 <x<π;

d.f(x)=|sinx|.

2. Sketch for at least two periods the graphs of the functions defined by:

a.f(x)=x, − 1 <x≤ 1 , f(x+ 2 )=f(x);