11.5.2.29 Fourier method

Problem: Solve the Dirichlet problem for the Laplace equation in the circle: \( \Delta u=0 \) in the circle \( 0 \leq r<1,0 \leq \varphi<2 \pi \), \[ u(1, \varphi)=5 \cos 5 \varphi \text {. } \]