Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Next Article
     

Euler's integrals and multiple sine functions

Author(s): Shin-ya Koyama; Nobushige Kurokawa
Journal: Proc. Amer. Math. Soc. 133 (2005), 1257-1265.
MSC (2000): Primary 11M06
Posted: December 15, 2004
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We show that Euler's famous integrals whose integrands contain the logarithm of the sine function are expressed via multiple sine functions.

References:

[E1]
L. Euler ``De summis serierum numeros Bernoullianos involventium'' Novi commentarii academiae scientiarum Petropolitanae 14 (1769) 129-167 [Opera Omnia I-15, pp.91-130].

[E2]
L. Euler ``Exercitationes analyticae'' Novi commentarii academiae scientiarum Petropolitanae 17 (1772) 173-204 [Opera Omnia I-15, pp.131-167].

[E3]
L. Euler ``Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques" Mémoires de l'académie des sciences de Berlin 17 (1761) 83-106 (Lu en 1749) [Opera Omnia I-15, pp. 70-90].

[H]
G.H. Hardy ``Divergent Series" Oxford Univ. Press 1949. MR 0030620 (11:25a)

[K1]
N. Kurokawa ``Multiple sine functions and Selberg zeta functions'' Proc. Japan Acad. 67A (1991) 61-64. MR 1105522 (92d:11094)

[K2]
N. Kurokawa ``Multiple zeta functions: an example'' Adv. Stud. in Pure Math. 21 (1992) 219-226. MR 1210791 (94f:11084)

[KK1]
N. Kurokawa and S. Koyama ``Multiple sine functions'' Forum Math. 15 (2003) 839-876. MR 2010282 (2004g:11077)

[KK2]
S. Koyama and N. Kurokawa ``Kummer's formula for multiple gamma functions'' J. Ramanujan Math. Soc. 18 (2003) 87-107. MR 1966530 (2004a:33003)

[KOW]
N. Kurokawa, H. Ochiai and M. Wakayama ``Multiple trigonometry and zeta functions'' J. Ramanujan Math. Soc. 17 (2002) 101-113. MR 1913896 (2003f:11132)

[KW]
N. Kurokawa and M. Wakayama ``On $\zeta(3)$'' J. Ramanujan Math. Soc. 16 (2001) 205-214. MR 1863604 (2002i:11083)

[M]
Yu. I. Manin ``Lectures on zeta functions and motives (according to Deninger and Kurokawa)'' Astérisque 228 (1995) 121-163. MR 1330931 (96d:11076)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11M06

Retrieve articles in all Journals with MSC (2000): 11M06

Additional Information:

Shin-ya Koyama
Affiliation: 2-5-27 Hayabuchi, Tsuzuki-ku, Yokohama 224-0025, Japan
Email: koyama@tmtv.ne.jp

Nobushige Kurokawa
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 152-8551, Tokyo, Japan
Email: kurokawa@math.titech.ac.jp

DOI: 10.1090/S0002-9939-04-07863-3
PII: S 0002-9939(04)07863-3
Received by editor(s): August 4, 2003
Posted: December 15, 2004
Communicated by: David E. Rohrlich
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google