Differentiation of functions of quaternionic variable

In this paper it is considered the definition of differentiability and regularity by Fueter [1, 2] and examples of regular function by Fueter, and the definition of C-regularity and C-derivative or Cullen derivative, on the basis of which a new theory of regular functions, which already includes polynomials and converging series of hypercomplex variable as differentiable and regular functions. Then a new definition of differentiability is proposed. It has a classical form, but specific convergence, which allows to prove theorems about differentiability of the sum and product of differentiable functions, differentiability of the “quotient” of differentiable functions. Further, it is deduced the derivative of power and is proved differentiability of polynomials and power series that allows to construct generalization of elementary functions for quaternionic argument. An example is given to show that without specific convergence the given definition of differentiability loses its meaning. With the help of power series functions are given, which are solutions of differential equations with constant quaternion coefficients. It is considered the problem of finding the roots of a square equation that arises in solving differential equations.