## Activity: Fourier Transform of a Derivative

Periodic Systems (2 years)
• This activity is used in the following sequences

The formula for the inverse Fourier transform shows that a function $f(x)$ can be written in terms of its Fourier transform via \begin{equation} f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \tilde{f}(k)\, e^{ikx}\, dk \end{equation} Take the derivative of both sides of this equation with respect to $x$ and simplify. Interpret your expression as the inverse Fourier transform of something.

## Instructor's Guide

### Introduction

Students will need a short lecture giving the definition of the inverse Fourier Transform \begin{equation} {\cal{F}}^{-1}(\tilde{f}) =f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(k)\, e^{ikx}\, dk \end{equation}

### Student Conversations

The logic of this problem may feel a little backwards to students. Be prepared to be more directive than normal in helping the groups that get stuck. Or consider doing this problem as a mini-lecture, rather than a group activity, especially if time is tight.

### Wrap-up

The result if this calculation is an essential formula in solving differential equations with Fourier transforms.

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