The Helmholtz equation is expressed as $$\nabla^2 \psi + \lambda \psi = 0$$. This equation occurs, for eg., after taking the Fourier transform (with respect to the time coordinate) of the wave equation in free space. While this equation for the case of $\lambda$ real is reasonably well discussed in the literature, the case where the coefficient $\lambda$ is complex-valued is not much discussed in the literature. However, such an equation does come up for the components of the electric and magnetic fields when we try to solve Maxwell's equations within a conductor. So my question is, what is the general solution of the equation $$\nabla^2 \psi + \lambda \psi = 0$$, where $\lambda$ is complex-valued(having both real and imaginary parts), in cylinderical coordinates? Also, can you quote a good reference from where I can get some good information about this?
[Physics] General solution to the Helmholtz wave equation with complex-valued frequency in cylinderical coordinates
classical-electrodynamicsdifferential equationselectromagnetismwaves
Best Answer
The so-called cylindrical waves can be chosen as the basis of the set of solutions (see, e.g., http://www.eecis.udel.edu/~weile/ELEG648Spring06/Resources/Cylindrical.pdf ). I guess the complex-valued $\lambda$ will lead to a complex-valued argument of the Bessel functions in the expressions for the cylindrical waves.