### Week 1 homework

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
, f(x+2)=f(x)
• c.
f(x)={ 0 -π
, f(x+3π)=f(x)
• d.
f(x)={ 0 -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
• 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
• 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.