Lecture Notes

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====== 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. $)

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