勒奇超越函数
外观
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lerch_transsendent.gif/220px-Lerch_transsendent.gif)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Lerch_complex_plot.gif/220px-Lerch_complex_plot.gif)
勒奇超越函数是一种特殊函数,推广了赫尔维茨ζ函数和多重对数函数,定义如下
特例
[编辑]- 赫尔维茨ζ函数。当勒奇函数中的z=1时,化为赫尔维茨ζ函数:
- 多重对数函数,当勒奇函数中a=1,则化为多重对数函数
- 勒让德χ函数可以用勒奇超越函数表示,
作为赫尔维茨ζ函数的特例,黎曼ζ函数可以表示为
狄利克雷η函数可以表示为
积分形式
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级数展开
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参考文献
[编辑]- Apostol, T. M., Lerch's Transcendent, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248.
- Bateman, H.; Erdélyi, A., Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill, 1953 [2015-02-14], (原始内容存档 (PDF)于2011-08-11). (See § 1.11, "The function Ψ(z,s,v)", p. 27)
- Gradshteyn, I.S.; Ryzhik, I.M., Tables of Integrals, Series, and Products 4th, New York: Academic Press, 1980, ISBN 0-12-294760-6. (see Chapter 9.55)
- Guillera, Jesus; Sondow, Jonathan, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, The Ramanujan Journal, 2008, 16 (3): 247–270, MR 2429900, arXiv:math.NT/0506319
, doi:10.1007/s11139-007-9102-0. (Includes various basic identities in the introduction.)
- Jackson, M., On Lerch's transcendent and the basic bilateral hypergeometric series 2ψ2, J. London Math. Soc., 1950, 25 (3): 189–196, MR 0036882, doi:10.1112/jlms/s1-25.3.189.
- Laurinčikas, Antanas; Garunkštis, Ramūnas, The Lerch zeta-function, Dordrecht: Kluwer Academic Publishers, 2002, ISBN 978-1-4020-1014-9, MR 1979048.
- Lerch, Mathias, Note sur la fonction , Acta Mathematica, 1887, 11 (1–4): 19–24, JFM 19.0438.01, MR 1554747, doi:10.1007/BF02612318 (法语).