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.