Generalized Exponential Function and some of its Applications to Complex Systems

Details Generalized Exponential Function and some of its Applications to Complex Systems Generalized Exponential Function and some of its Applications to Complex Systems

2008-12-16

arxiv.org/abs/0812.3071v1

Advances in Mathematical Physics, v. 2009, p. 1-14, 2009

Physics

Data Analysis, Statistics and Probability

16 pages and 2 figures

Scientific paper

From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing complex systems can be conveniently written in terms of this generalization of the exponential function. The gamma function is then generalized and we generalize the factorial operation. Also a very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. One obtains, as particular cases, the generalized error function, the Zipf-Mandelbrot probability density function (pdf), the generalized gaussian and Laplace pdf. One can also obtain analytically their cumulative functions and moments.

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