Section 1.1

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
e^x = cosh(x) + sinh(x)
gives the decomposition of e^x 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.