We consider the heat equation: $\frac{\partial f}{\partial r}(r,x)=\frac{\partial^2 f}{\partial x^2}(r,x),(r,x) \in \mathbb{R}^*_+ \times \mathbb{R}.$
We want to find a solution of this equation using Fourier series.
I know that $p(r,x)=\sum_{k \in \mathbb{Z}}e^{2 \pi i k x}e^{-(2 \pi k)^2 r},(r,x) \in \mathbb{R}^*_+ \times \mathbb{R}$ is a solution, and we can prove it by showing that $p$ verifies the equation.
Given the equation $\frac{\partial f}{\partial r}(r,x)=\frac{\partial^2 f}{\partial x^2}(r,x),$ how to obtain the expression of $p$?