Lerch’s transcendent, named after Czech mathematician
Mathias Lerch (1860 – 1922) is defined by the power series :
Where a ≠ 0, -1, -2, … on the domain |z| < 1 for any s ∈ ℂ or |z| ≤ 1 for ℝ > 1 .
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 ,
- Representation of distributions in particle physics. For example, in the Bose-Einstein Distribution (which describes the statistical behavior of bosons) .
- Production of urban noise maps .
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 . 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 . Lerch’s transcendent generalizes the Hurwitz zeta function reducing to the Hurwitz zeta function when z = 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.
 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
 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.
 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
 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
 Navas, L. (2015). The Lerch transcendent from the point of view of Fourier analysis. Journal of Mathematical Analysis and Applications.
 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.
Stephanie Glen. "Lerch’s Transcendent" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/lerchs-transcendent/
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