We’ve gone over Heat Equation, Wave Equation, and let’s talk about some more stuff.

• damped heat equation
• damped wave equation
• two-dimensional heat equation

What if, Fourier Series, but exponential?

This also motivates Discrete Fourier Transform.

Also Complex Exponential.

Review

Recall again that if we have a periodic function, we’ve got:

\begin{equation} f(x) = \sum_{k=0}^{\infty} a_{k} \sin \qty( \frac{2\pi k}{l} x) + b_{n} \cos \qty( \frac{2\pi k x}{L}) \end{equation}

We note that this breaks individually into the sign and cosine series depending of the function’s oddness.

Complex Fourier Series

This will begin by feeling like a notation rewrite:

Fourier Transform