Principles of Fourier Analysis, 2nd Edition

38

Pole Functions and General Solutions to Simple Equations

This chapter is about division. More precisely, it is about solving equations of the form fu = g for u. To see why we might want to spend an entire chapter on this, let’s look at some problems that arise when using Fourier transforms to solve the differential equations

$\frac{dy}{dt}+3y=\delta \text{and}\frac{d^{2}y}{d{t}^2}.$

Taking the transform of both sides of the first equation and using a differentiation identity yields

$i2\pi \omega Y\left(\omega \right)+3Y\left(\omega \right)=1$

where $Y=\mathcal{F}\left[y\right]$. By elementary algebra, this can be written as

$\left[3+i2\pi \omega \right]Y\left(\omega \right)=1.$

Dividing through by 3 + i2πω then gives

$Y\left(\omega \right)=\frac{1}{3+i2\pi \omega },$

and from this we obtain

$y\left(t\right)={\mathcal{F}}^{-1}{\left[\frac{1}{3+i2\pi \omega }\right]|}_t=e^{-3t}\text{step}\left(t\right).$

You can easily verify that this is a solution to the differential equation. ...