## L-Periodic Functions

So, we have:

\begin{equation} f(x+L) = f(x) \end{equation}

The integral is equivalent for any:

\begin{equation} \int_{a}^{a+L} f(x) \end{equation}

for any \(a\).

## Heat Equation Recipe

- are we on a finite interval? then, decompose into product-type solution \(A(t)B(x)\) and solve.
- are we not? Fourier transform on the space variable and solve.

### What if \(\lambda \in \mathbb{C} \backslash \mathbb{R}\)

Shush.

### Why can we guess \(A(t)B(x)\)

Because we were able to find solutions. Believe that the solution set spans.

## Fourier Transform on Three-Variable Expressions

We have better Fourier transforms on n-space rather than on a line. Use those.