## Small Group Activity: Fourier Transform of a Shifted Function

Periodic Systems 2022
• This ingredient is used in the following sequences

Suppose you have a definite function $f(x)$ in mind and you already know its Fourier transform, i.e. you know how to do the integral $$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}e^{-ikx}\, f(x)\, dx$$ Find the Fourier transform of the shifted function $f(x-x_0)$.

## Instructor's Guide

### Introduction

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

### Student Conversations

This example will feel very abstract to some students. It may be difficult for them to understand that the conditions of the problem state that the know both $f(x)$ and $\tilde{f}(k)$. This problem is about changing $f$ slightly (by shifting its argument by $x_0$) and then asking how $\tilde{f}$ changes, in response.

### Wrap-up

The result from this calculation underlies why it is possible to factor out the time dependence in the Fourier transform of a plane wave, Fourier Transform of a Plane Wave. Even though the problem is somewhat abstract, it is super important in applications for this reason.

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