Homework 6

Due on Monday (Oct 9th)

1. Find the Fourier transformation of the following function. $$ f(x) = \begin{cases} 1 & 0<x<1 \cr 0 & \text{else} \end{cases} $$

2. Find the Fourier transformation of the following function. $$ f(x) = \begin{cases} 1+x & -1<x<0 \cr 1-x & 0<x<1 \cr 0 & \text{else} \end{cases} $$

3. Let $f(x) = a / (x - i) + b / (x+i) + c / (x - 5i)$ for some complex numbers $a,b, c$.

• Describe for what choices of $a,b,c$ the function $f(x)$ is absolutely integrable.
• Take $a=2, b=-1, c=-1$, where we know $f(x)$ is integrable, compute its Fourier transformation.

4. Compute the inverse Fourier transform for $$ F(p) = \pi e^{-|p|} $$ you should get back $f(x) = 1/(1+x^2)$.

(my convention is $f(x) = \frac{1}{2\pi} \int F(p) e^{ipx} dp. $)