- 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.

- 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)=x
^{2}, -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.