勒奇超越函数:修订间差异
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==参考文献== |
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* {{dlmf | id= 25.14 | first= T. M. | last= Apostol | title= Lerch's Transcendent}}. |
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* {{citation | first1= H. | last1= Bateman | author1-link= Harry Bateman | first2= A. | last2= Erdélyi | author2-link= Arthur Erdélyi | title= Higher Transcendental Functions, Vol. I | year= 1953 | location= New York | publisher= McGraw-Hill | url=http://apps.nrbook.com/bateman/Vol1.pdf}}. (See § 1.11, "The function Ψ(''z'',''s'',''v'')", p. 27) |
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* {{citation | first1= I.S. | last1= Gradshteyn | first2= I.M. | last2= Ryzhik | title= Tables of Integrals, Series, and Products | edition= 4th | location= New York | publisher= Academic Press | year= 1980 | isbn= 0-12-294760-6}}. (see Chapter 9.55) |
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* {{citation | first1= Jesus | last1= Guillera | first2= Jonathan | last2= Sondow | arxiv= math.NT/0506319 | mr = 2429900 | title= Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent | journal= The Ramanujan Journal | volume= 16 | year= 2008 | pages= 247–270 | issue= 3 | doi= 10.1007/s11139-007-9102-0}}. (Includes various basic identities in the introduction.) |
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* {{citation | first= M. | last= Jackson | title= On Lerch's transcendent and the basic bilateral hypergeometric series <sub>2</sub>''ψ''<sub>2</sub> | year= 1950 | journal= J. London Math. Soc. | volume= 25 | issue= 3 | pages= 189–196 | doi= 10.1112/jlms/s1-25.3.189 | mr= 0036882}}. |
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* {{citation | first1= Antanas | last1= Laurinčikas | first2= Ramūnas | last2= Garunkštis | title= The Lerch zeta-function | publisher= Kluwer Academic Publishers | location= Dordrecht | year= 2002 | isbn= 978-1-4020-1014-9 | mr= 1979048}}. |
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* {{citation | first= Mathias | last= Lerch | authorlink= Mathias Lerch | title= Note sur la fonction <math>\scriptstyle{\mathfrak K}(w,x,s) = \sum_{k=0}^\infty {e^{2k\pi ix} \over (w+k)^s}</math> | language= French | year= 1887 | journal= Acta Mathematica | volume= 11 | issue= 1–4 | pages= 19–24 | doi= 10.1007/BF02612318 | mr= 1554747 | jfm= 19.0438.01}}. |
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[[Category:特殊函数]] |
2015年2月14日 (六) 02:50的版本
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勒奇函数是一种特殊函数,定义如下
特例
- 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
积分形式
级数展开
参考文献
References
- 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. (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 (French) .