Full of Math Examples
FULL OF MATH EXAMPLES

# Fourier Transform Examples

Here we will learn about Fourier transform with examples.

## Definition of Fourier Transform

The Fourier transform of $f(x)$ is denoted by $\mathscr{F}\{f(x)\}= $$F(k), k \in \mathbb{R}, and defined by the integral : \mathscr{F}\{f(x)\}=F(k)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-i k x} f(x) d x Where \mathscr{F} is called fourier transform operator. The inverse Fourier transform, denoted by \mathscr{F}^{-1}\{F(k)\}=f(x), is defined by \mathscr{F}^{-1}\{F(k)\}=f(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{i k x} F(k) d k where \mathscr{F}^{-1} is called the inverse Fourier transform operator. ## Examples of Fourier Transform ### Example #1 #### Find the Fourier transform of exp \left(-a x^{2}\right) . By fourier transform formula we have, F(k)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-i k x-a x^{2}} d x =\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \exp \left[-a\left(x+\frac{i k}{2 a}\right)^{2}-\frac{k^{2}}{4 a}\right] d x =\frac{1}{\sqrt{2 \pi}} \exp \left(-k^{2} / 4 a\right) \int_{-\infty}^{\infty} e^{-a y^{2}} d y =\frac{1}{\sqrt{2 a}} \exp \left(-\frac{k^{2}}{4 a}\right) Here is the graph of fourier transform ### Example #2 #### Find the Fourier transform of a below non periodic function f(x)=\left\{\begin{array}{ll}{1,} & {-1\lt x\lt 1} \\ {0,} & {|x|\gt 1}\end{array}\right. The above function is not a periodic function. A non periodic function cannot be represented as fourier series.But can be represented as Fourier integral. Then,using Fourier integral formula we get, F(k)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} f(x) e^{-i k x} d x =\frac{1}{2 \pi} \int_{-1}^{1} e^{-i k x} d x =\frac{1}{2 \pi}\left.\frac{e^{-i k x}}{-i k}\right|_{-1} ^{1}$$ =\frac{1}{\pi k} \frac{e^{-i k}-e^{i k}}{-2 i}$$=\frac{\sin k}{\pi k}$

This is the Fourier transform of above function.

We can find Fourier integral representation of above function using fourier inverse transform.

$f(x)=\int_{-\infty}^{\infty} \frac{\sin k}{\pi k} e^{i k x} d x$
$=\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\sin k(\cos k x+i \sin k x}{k} d k$
$=\frac{2}{\pi} \int_{0}^{\infty} \frac{\sin k \cos k x}{k} d k$

This is the fourier integral representation of our non periodic function.