Functions of a quaternion variable which are gradients of real-valued functions (Q1082491)

We consider the collection of functions of one quaternion variable which can be expressed as G(Y) where Y is a real-valued quaternion function and G is a differential operator which corresponds to the gradient of real variable theory. Integral theorems for such functions are given, together with necessary and sufficient conditions for a function to be a gradient function, in terms of its Fréchet derivative. The extended complex analytic functions, the Fueter functions, and the momentum-energy density functions are seen to be gradient functions which correspond to biharmonic, harmonic, and wave functions respectively.