Analyticity of the Complex Time Scale Exponential

Abstract

In this paper, we prove the analyticity property of the complex time scales exponential function $z\mapsto \mathrm {e}_{z}(t,s)$, where $s,t\in \mathbb {T}$ and z is a regressive complex variable. We also prove various properties of the function $z\mapsto \int _{s}^{t}\frac{1}{1+z\mu (\eta )}\Delta \eta$, where $s,t\in \mathbb {T}$, one of which yields the alternative expression $\mathrm {e}_{z}(t,s)=\exp \big \{\int _{0}^{z}\int _{s}^{t}\frac{1}{1+\zeta \mu (\eta )}\Delta \eta \mathrm {d}\zeta \big \}$ of the exponential function in terms of the contour integral in the regressive complex domain.