Analyticity of the Complex Time Scale Exponential

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• 作者：Başak Karpuz
• 关键词：Analytic functions ; Differential equations in the complex plane ; Time scales exponential function
• 刊名：Complex Analysis and Operator Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：11
• 期：1
• 页码：21-34
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Operator Theory; Analysis;
• 出版者：Springer International Publishing
• ISSN：1661-8262
• 卷排序：11

文摘

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.