The Mittag–Leffler (ML) relaxation function, E α (− t α ) (0 < α ⩽ 1), describes multiscale relaxation processes with a broad range of relaxation rates, where α = 1 corresponds to exponential relaxation. For 0 < α < 1 it decays asymptotically ∼ t − α and is thus asymptotically self-similar, i.e. form invariant under a scale transform t → μt . In the language of asymptotic analysis, such functions are referred to as regularly varying. Based on this observation we derive a refined, ‘weakly self-similar’ asymptotic form by applying a theorem due to J Karamata. Reasoning along the same lines, we derive also a corresponding weakly self-similar form for the time derivatives of the ML relaxation function in the short time limit. In both cases the respective asymptotic power law forms are approached by slowly varying functions in the sense of asymptotic analysis and we s...

中文翻译：

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Mittag-Leffler松弛函数的弱自相似性

Mittag-Leffler（ML）松弛函数Eα（-tα）（0 <α⩽1）描述了具有宽范围松弛率的多尺度松弛过程，其中α= 1对应于指数松弛。当0 <α<1时，它渐近衰减到t-α，因此是渐近自相似的，即在尺度转换t→μt下形式不变。用渐近分析的语言，这些功能被称为规则变化。基于此观察，我们通过应用J Karamata的一个定理，推导出一种精炼的“弱自相似”渐近形式。沿着相同的路线进行推理，我们还为短时限内的ML松弛函数的时间导数推导了相应的弱自相似形式。