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Lerch’s Transcendent

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Lerch’s transcendent, named after Czech mathematician
Mathias Lerch (1860 – 1922) is defined by the power series [1]:
lerch's transcendent

Where a ≠ 0, -1, -2, … on the domain |z| < 1 for any s ∈ ℂ or |z| ≤ 1 for ℝ > 1 [2].

Where Lerch’s Transcendent is Used

Many sums of reciprocal powers can be expressed in terms of Lerch’s transcendent function; It makes many appearances in physical science problems. For example:


  • Formulation of electrostatic problems [3],
  • Representation of distributions in particle physics. For example, in the Bose-Einstein Distribution (which describes the statistical behavior of bosons) [4].
  • Production of urban noise maps [5].

In number theory, Lerch’s transcendent provides a unified framework for the study of many special functions. It can be obtained from the closely related Lerch zeta function by a change of variable z = e2πia [6]. Lerch’s functions are usually of interest because their analytic continuations include, as special cases, several important transcendental functions including the the polylogarithm function and the Riemann zeta function [7]. Lerch’s transcendent generalizes the Hurwitz zeta function reducing to the Hurwitz zeta function when z = 1.

References

[1] Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953a) Higher Transcendental Functions. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.
[2] Cai, X. & Lopez, J. (2021). A note on the asymptotic expansion of the Lerch’s transcendent. Retrieved April 24, 2021 from: http://arxiv-export-lb.library.cornell.edu/pdf/1806.01122
[3] Ivchenko, V. (2020). On the interaction force between a point charge and an infinite dielectric plate of finite thickness. European Journal of Physics, Volume 41, Issue 1, id.015201.
[4] Awan, A. (2015). On the Theory of Zeta-functions and L-functions. Retrieved April 24, 2021 from: https://stars.library.ucf.edu/cgi/viewcontent.cgi?article=1052&context=etd
[5] Lagarias, J. & Li, W. The Lerch zeta function III. Polylogarithms and special values. Retrieved April 24, 2021 from: https://ui.adsabs.harvard.edu/abs/2015arXiv150606161L/abstract
[6] Navas, L. (2015). The Lerch transcendent from the point of view of Fourier analysis. Journal of Mathematical Analysis and Applications.
[7] Wei, W. et al. (2015). Simplified analytical model for sound level prediction at shielded urban locations involving multiple diffraction and reflections. J Acoust Soc Am Nov;138(5):2744-58. doi: 10.1121/1.4932585.


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Stephanie Glen. "Lerch’s Transcendent" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/lerchs-transcendent/
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