Hi friends. I'm trying to solve a BVP by using Fourier series. It is an electormagnetic scattering problem from an infinite cylinder whose surface impedance is known. Cylinder is illuminated by a line source. After taking fourier transform with respect to phi, scalar helmholtz equation reduced to bessel diff. equation.

Eventually, three equations have arised and there are three unknown An, Bn, Cn. First two equations are easy to handle and I have no problem with them. One of them is:

Third-problematic- equation is:

Z(phi) is surface impedance of the cylinder. Actaully we know it(Dm) but we behave as if it is not known to find a general solution. Multiplication of infinite series(fourier series) confuses me. How can we derive an equation like first two from third equation? Does orthonality principle work here? Any idea-comment will be welcome. Thanks.