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勒奇超越函數

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勒奇超越函数
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勒奇超越函數是一種特殊函數,推廣了赫爾維茨ζ函數多重對數函數,定義如下

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Lerch transcendent
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Lerch plot with complex variable


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特例

赫爾維茨ζ函數。當勒奇函數中的z=1時,化為赫爾維茨ζ函數:

多重對數函數,當勒奇函數中a=1,則化為多重對數函數
勒讓德χ函數可以用勒奇超越函數表示,

作為赫爾維茨ζ函數的特例,黎曼ζ函數可以表示為

狄利克雷η函數可以表示為

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積分形式

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級數展開


參考文獻

  • Apostol, T. M., Lerch's Transcendent, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), 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], (原始內容存檔 (PDF)於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可免費查閱, 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 (法語).
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