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

We consider the collection of functions of one quaternion variable which can be expressed asG(Y) whereY is a real-valued quaternion function andG 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 Frechet 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.