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 and 1, 0<x<π
d. f(x)=|sin(x)|
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)
b.
f(x)={
0
-1<x<=0
x
0<x<1
, f(x+2)=f(x)
c.
f(x)={
0
-π<x<=0
1
0<x<=2π
, f(x+3π)=f(x)
d.
f(x)={
0
-π<x<=0
sin x
0<x<=π
, f(x+2π)=f(x)
3. Show that the constant function f(x)=1 is periodic with every
possible period p>0.
Section 1.2
1. Find the Fourier series of each of the following functions.
Sketch the graph of the periodic extension of f for at least two periods.
a. f(x)=|x|, -1<x<1
b.
f(x)={
-1
-2<x<0
1
0<x<2
c. f(x)=x2, -1/2 < x < 1/2
2. Show that the functions cos(nπx/a) and sin(nπx/a)
satisfy orthogonality relations similar to those given in Section 1.
4. Show that the formula
ex = cosh(x) + sinh(x)
gives the decomposition of ex into a sum of an odd
function and an even function.
5. Identify each of the following as being even, odd, or neither. Sketch.
a. f(x)=x
b. f(x)=|x|
c. f(x)=|cos(x)|
d. f(x)=arcsin(x)
e. f(x)=x cos(x)
f. f(x)=x+cos(x+1)
7. Find the Fourier series of the functions
a. f(x) = x, -1<x<1
b. f(x) = 1, -2<x<2
c.
f(x)={
x
-1/2<x<1/2
1-x
1/2<x<3/2
10. Sketch both the even and odd extensions of the functions:
a. f(x)=1, 0<x<a
b. f(x)=x, 0<x<a
c. f(x)=sin(x), 0<x<1
d. f(x)=sin(x), 0<x<π
11. Find the Fourier sine series and cosine series for the
functions given in Exercise 10. Sketch the even and odd
periodic extensons for several periods.