Fourier analysis problem (problem number 5)
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Chapter 1. THE GENESIS OF FOURIER ANA
5. Verify that f(x) = eins is periodic with period 27 and that
了。 cim ds = {
1 if n = 0,
0 if n +0.
Use this fact to prove that if n, m > 1 we have
so if n #m,
cos na cos mu dr =
1 n = m,
7T
-
and similarly
1
S.
sin nx sin mu dx =
{
0 if n #m,
1 n = m.
Finally, show that
L
sin nx cos mx dx = 0
for any n,m.
(Hint: Calculate einte-imz + einzeime and ente -ima
eina ima]
6. Prove that if f is a twice continuously differentiable function on R
a solution of the equation
f"(t) + f(t) = 0,
then
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