Axial Monogenic Functions from Holomorphic Functions

Monogenic functions with axial symmetry are closely related to holomorphic functions of a single complex variable and recently generalisations of the exponential function, power functions, and Hermite polynomials have been discussed in some detail. We present here an explicit integral transform of a large class of holomorphic functions which gives a corresponding class of axial monogenic functions. ft is demonstrated that these integral transforms are related to plane wave monogenic functions. The axial monogenic power and exponential functions are constructed from powers and exponential functions of a complex variable and a new bound on the modulus of the axially monogenic exponential function is derived. Finally monogenic generating functions are employed to define generalisations of Gegenbauer polynomials as well as recreating generalised Hermite polynomials.