Let m and p be constants, and let u$($x$,$ y$)$ be a function whose derivatives up to the third order are continuous. Consider an H field defined in the Oxyz Cartesian coordinate system as:

Hx $=-$p$($u$/$y$)$ e$^($ipz$),$ Hy $=$ i$($u$/$x$)$ e$^($ipz$),$ Hz $=0.$

a$)$ Find the homogeneous differential equation that must be satisfied by p and the function u$($x$,$ y$)$ for H to be a monochromatic magnetic field propagating in a simple medium with constants and $,$ no conductivity, and no sources.

b$)$ If m and u$($x$,$ y$)$ satisfy the above properties, write E$.$

c$)$ Calculate the complex Poynting vector.

Explain step by step.