EXERCISES 3.2

3.2.1. For the following functions, sketch the Fourier series of f(x) (on the interval
-L < x < L). Compare f (x) to its Fourier series:

(a) f(x) = 1    *(b) f(x) = x^2
(c) f(x)=1+x    *(d) f(x) = e^x
(e) f(x) = x for x<0 and f(x) =2x for x>0
*(f) f(x)=1+x for x>0 and zero otherwise
(g) f(x)=x for x<L/2 and zero otherwise


3.2.2. For the following functions, sketch the Fourier series of f(x) (on the interval
-L < x < L) and determine the Fourier coefficients:

*(a) f(x)=x              (b) f(x) = e^{-x}
*(c) f(x) = sin(Pi x/L)  (d) f(x)=x for x>0 and zero otherwise
(e) f(x)=1 for |x|<L/2 and zero otherwise
*(f) f(x)=1 for x>0 and zero otherwise
(g) f(x)=1 for x<0 and f(x)=2 for x>0


3.2.3. Show that the Fourier series operation is linear: that is, show that the
Fourier series of c_1 f(x) + c_2 g(x) is the sum of c_1 times the Fourier series of
f(x) and c_2 times the Fourier series of g(x).


3.2.4. Suppose that f(x) is piecewise smooth. What value does the Fourier series
of f(x) converge to at the endpoint x = -L? at x = L?