Complex generalized exponential function for real argument

Since their introduction by Tsallis in the late 1980s, generalized functions have been widely studied and used in different areas of knowledge. However, the non-analyticity in the complex plane of these functions has been an obstacle to expand their applicability. In this context, we present here a brief review of the generalized logarithm and exponential functions and propose a generalization for the latter that makes it analytic in the complex plane. This new generalization opens possibilities for new applications and extensions of this theory.

Keywords:
Generalized exponential function in complex plane; Generalized logarithm function; Analytic exponential function in complex plane; Solution of cubic equations