# Homework #6

## 4 thoughts on “Homework #6”

1. Question from a student:

For question 1B, which theorem should I use?

You are meant to (somehow) use the inversion formula:
$f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty\widehat{f}(\omega)e^{+i\omega t}\,d\omega$

We’ve been writing the inversion formula since we first talked about Fourier transform 30-Jan. What’s new ( and theorem ) is that you can actually trust the inversion formula to tell the truth.

See 06-Feb, slide 5

2. Question from a student:

For question 2B, partial fraction clearly won’t work on this. And if I complete the square, the equation looks horrible. How should I tackle this question?

You have the right idea – you just need to stick with it.

You can factor the denominator as:
$\widehat{g}(\omega) = \frac{1}{(i\omega+A)(i\omega+B)}$,
with constants $A$ and $B$ not too horrible. Then use partial fractions.

• I still don’t get it. If coefficients of both “iw” are 1 then one of the terms we get is $-w^2$ but we want $-2w^2$. So far I have done this $\frac{1}{ (1+iw)^2 - w^2 }$

• You need to factor out the 2. eg: $\frac{1}{2}\,\frac{1}{-\omega^2+i\omega+\frac{1}{2}}$