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首页> 外文期刊>Journal of Non-Crystalline Solids: A Journal Devoted to Oxide, Halide, Chalcogenide and Metallic Glasses, Amorphous Semiconductors, Non-Crystalline Films, Glass-Ceramics and Glassy Composites >Is there geometrical/physical meaning of the fractional integral with complex exponent?
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Is there geometrical/physical meaning of the fractional integral with complex exponent?

机译：具有复数指数的分数积分有几何/物理意义吗？

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摘要

The geometrical/physical meaning of the temporal fractional integral with complex fractional exponent has been found and discussed. It has been shown that the imaginary part of the fractional integral is related to discrete scale invariance (DSI) phenomenon and observed only for trite regular (discrete) fractals. Numerical experiments show that the imaginary part of the complex fractional exponent can be well approximated by a simple and finite combination of the leading sine/cosine log-periodical functions with period In xi (xi is a scaling parameter). In most cases analyzed, the leading Fourier components give a pair of complex conjugated exponents defining the imaginary part of the complex fractional integral. For random fractals, where invariant scaling properties are realized only in the statistical sense the imaginary part of the complex exponent is averaged and the result is expressed in the form of the conventional Riemann-Liouville integral. The conditions for realization of reind and recaps elements with complex power-law exponents have been found. Description of relaxation processes by kinetic equations containing complex fractional exponent and their possible recognition in the dielectric spectroscopy is discussed. New kinetics expressed in terms of non-integer operators with complex and real power-law exponents can be successfully applied for description of dielectric spectra of many non-crystalline solids. (c) 2005 Elsevier B.V. All rights reserved.
机译：已经发现并讨论了具有复杂分数指数的时间分数积分的几何/物理意义。已经表明，分数积分的虚部与离散尺度不变性（DSI）现象有关，并且仅在三重规则（离散）分形中观察到。数值实验表明，复杂的小数指数的虚部可以通过将正弦/余弦对数周期函数与周期xi xi（xi是缩放参数）进行简单而有限的组合而很好地近似。在大多数情况下，前导傅里叶分量给出一对复共轭指数，它们定义了复数分数积分的虚部。对于仅在统计意义上实现不变缩放属性的随机分形，将复指数的虚部平均，并以传统的黎曼-利维尔积分形式表示结果。已经找到了实现具有复杂幂律指数的reind和recaps元素的条件。讨论了通过包含复数分数指数的动力学方程描述弛豫过程的方法，以及在介电谱中可能的识别方法。用具有复杂和实数幂律指数的非整数算子表示的新动力学可以成功地用于描述许多非晶态固体的介电谱。 （c）2005 Elsevier B.V.保留所有权利。

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