Maxwell's equations in bicomplex (quaternion) form: an alternative to the Helmholtz P.D.E

The concept of bicomplex numbers is introduced in electromagnetics, with direct application to the solution of Maxwell's equations. It is shown that, with the assistance of a bicomplex vector field, defined as a combination of the electric and the magnetic fields, the number of unknown quantities is practically reduced by half, whereas the Helmholtz equation may no longer be necessary in the development of the final solution. Bicomplex first order differential equations are involved, instead of conventional complex second order equations, and the solution procedure is greatly simplified. A direct consequence of this observation is the derivation of closed form solutions of the Maxwell's equations for a special class of inhomogeneous media, which cannot be easily extracted from the Helmholtz equation alone.