Electromagnetic Boundary Problems

6    Propagation and Scattering in More Complex Regions

6.1    GENERAL CONSIDERATIONS

It is worth asking at this point what it is about the transform representations (Fourier series and Fourier transform) that makes them so useful in obtaining solutions for the fields in various geometrical arrangements of matter. Some consideration shows that the main reason is that these transforms convert a part of the relevant differential operators into algebraic multipliers. For example, in a Fourier series,

$\frac{{\partial }^2}{\partial {\varphi }^2}\left\{\begin{array}{c}\sin m\varphi \\ \cos m\varphi \\ e^{jm\varphi }\end{array}\right\}=-m^2\left\{\begin{array}{c}\sin m\varphi \\ \cos m\varphi \\ e^{jm\varphi }\end{array}\right\}$

If in addition the trigonometric or complex exponential functions obey whatever boundary or periodicity conditions are appropriate to our problem, we say that they are eigenfunctions of the operator ...