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Based on the full similarity in algebraic properties and differentiation rules between quaternionic (H-) holomorphic and complex (C-) holomorphic functions, we assume that there exists one holistic notion of a holomorphic function that has a H-representation in the case of quaternions and a C-representation in the case of complex variables. We get the essential definitions and criteria for a quaternionic power series convergence, adapting complex analogues to the quaternion case. It is established that the power series expansions of any holomorphic function in C- and H-representations are similar and converge with identical convergence radiuses. We define a H-analytic function and prove that every H-holomorphic function is H-analytic. Some examples of power series expansions are given., Comment: 22 pages

A physically more adequate definition of a quaternionic holomorphic (H-holomorphic) function of one quaternionic variable compared to known ones and a quaternionic generalization of Cauchy-Riemann's equations are presented. At that a class of introduced H-holomorphic functions consists of those quaternionic functions whose left and right derivatives become equal after the transition to 3D space. The presented theory demonstrates a complete similarity of the algebraic properties and differentiation rules between the classes of H-holomorphic and ordinary complex holomorphic functions, including the fact that quaternionic multiplication of the H-holomorphic functions behaves as commutative and the fact that each H-holomorphic function can be created from its complex holomorphic analogue by replacing a complex variable by a quaternion one. A fairly large number of detailed examples are given to illustrate the presented theory efficiency., Comment: 16 pages