I had to solve $\Delta u=0$ in the semi infinite strip $S=\{(x,y) |0<x<1,0<y \}$ with boundaries condition $$ u(x,0)=f(x)=\sum_{n\in\mathbb Z}a_n \sin(\pi n x), ~u(0,y)=u(1,y)=0 $$ then it gaves another condition that leads to the solution $u_n(x,y)=a_ne^{-\pi n y}\sin(\pi nx)$, and then to the general solution $$ u(x,y)=\sum_{n\in\mathbb Z}a_ne^{-\pi n y}\sin(\pi nx) $$
Then the exercise asks to write $u(x,y)$ as a integral involving $f$, like the Poisson integral formula $$ u(r,\theta)=f*P_r(\theta), $$ $P_r$ the Poisson kernel.
This last part is the one that I can't solve.
Thanks in advance
Any integral formulation comes from the explicit use of the Fourier coefficients: $$ a_n = 2\int_{0}^{1}f(u)\sin(n\pi u)du $$ Then $$ u(x,y)=2\sum_{n=1}^{\infty}\int_{0}^{1}f(u)\sin(n\pi u)du\sin(n\pi x)e^{-n\pi y} \\ = \int_{0}^{1}f(u)\left[2\sum_{n=1}^{\infty}\sin(n\pi u)\sin(n\pi x)e^{-n\pi y}\right]du $$