Fourier Equations Homework Solutions

[Invyz]

Homework Statement

Attached.

Homework Equations

The Attempt at a Solution

For #1: I've attempted to use the identity sin(x)sin(y) = 1/2(cos(x-y)-cos(x+y)) and the similar related identities for sin(x)cos(y) and cos(x)cos(y), but I end with answers that are undefined when m = n due to the term (m-n).

For #2: I don't see any derivation in the lecture notes(attached), so I don't know where to start :/ Other than that you probably begin with the summation as n goes to infinity of the Fourier constant times the cosine(2*pi*n*t/T)

 

[clamtrox]

For #1: I've attempted to use the identity sin(x)sin(y) = 1/2(cos(x-y)-cos(x+y)) and the similar related identities for sin(x)cos(y) and cos(x)cos(y), but I end with answers that are undefined when m = n due to the term (m-n).

Do the case m=n separately. You can probably calculate the integral by hand in that case.

For #2: I don't see any derivation in the lecture notes(attached), so I don't know where to start :/ Other than that you probably begin with the summation as n goes to infinity of the Fourier constant times the cosine(2*pi*n*t/T)

Write function f using the Fourier transform, multiply both sides of the equation by cos(2πmx/L). Then integrate both sides using the results you got from #1.