Fourier Transforms

Chapter 7.12 in "Mathematical Methods in the Physical Sciences" 3e by Mary L. Boas.

These are so cool. Go check out the Falstad Fourier applet.

Fourier series are useful for visualizing periodic functions as a sum of discrete coefficients and sines/cosines. $$\begin{array}{rl}f(x)& =\sum _{-\infty }^{\infty }{c}_n{e}^{in\pi x/l}\\ c_n& =\frac{1}{2l}{\int }_{-l}^{l}f(x)e^{-in\pi x/l}dx\end{array}$$

Visually, this GIF from the Wikipedia article on Fourier series is delightful.

Fourier integrals on the other hand are useful for representing both nonperiodic and periodic functions in terms of sines and cosines.

Let $f(x)$ be the function and $g(\alpha )$ its Fourier coefficients. The associated Fourier transforms are $$f(x)={\int }_{-\infty }^{\infty }g(\alpha )e^{i\alpha x}d\alpha$$ $$g(\alpha )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }f(x)e^{-i\alpha x}dx$$

Though these are in terms of $e^{-i\alpha x}$, we can convert them using Euler's formula: $$e^{-i\alpha x}=\cos (\alpha x)-i\sin (\alpha x)$$

To solve them,

1. Plug the original function $f(x)$ into the $g(\alpha )$ equation.
2. Solve for $g(\alpha )$.
3. Solve for the $f(x)$ Fourier transform.

Cosine and sine transforms

We can use cosine and sine transformations to represent even and odd functions respectively to simplify the calculation of Fourier integrals.

Cosine Transform: if $f(x)$ is an even function where $f(x)=f(-x)$, $$\begin{array}{rl}f(x)& =\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }g(\alpha )\cos \alpha xd\alpha \\ g(\alpha )& =\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(x)\cos \alpha xdx\end{array}$$

Sine Transform: if $f(x)$ is an odd function where $f(x)=-f(-x)$, $$\begin{array}{rl}f(x)& =\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }g(\alpha )\sin \alpha xd\alpha \\ g(\alpha )& =\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(x)\sin \alpha xdx\end{array}$$