The dilogarithm function for complex argument

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 459, Issue 2039, 2003, Pages 2807-2819

  Maximon, Leonard C ^a  

^a GEORGE WASHINGTON UNIVERSITY   (United States)

Author keywords

Clausen's integral; Debye function; Euler dilogarithm; Jonqui re's function; Polylogarithms; Spence function

Indexed keywords

ALGORITHM; ARTICLE; GEOMETRY; MATHEMATICS; PHYSICS; THEORY;

EID: 1542407751     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2003.1156     Document Type: Article

References (39)

1. 2 + ⋯. In Oeuvres complètes de Niels Henrik Abel, Nouvelle édition publée aux frais de l'Etat Norvégien, par MM. L. Sylow et S. Lie, vol. 2, pp. 189-193. Christiana: Grondahl & Son. (Reprinted by Johnson Reprint Corp, New York, 1965.)
2. Abramowitz, M. & Stegun, I. A. (eds) 1972 Handbook of mathematical functions with formulas, graphs and mathematical tables, 10th edn. National Bureau of Standards in Applied Mathematics, series 55. Washington, DC: US Government Printing Office.
3. Andrews, G. E., Askey, R. & Roy, R. 1999 Special functions. In Encyclopedia of mathematics and its applications, vol. 71. Cambridge University Press.
4. Barbieri, R., Mignaco, J. A. & Remiddi, E. 1971 Fourth-order radiative corrections to electron-photon vertex and the Lamb-shift value. Nuovo Cim. A 6, 21-28.
5. Barbieri, R., Mignaco, J. A. & Remiddi, E. 1972a Electron form factors up to fourth order. I. Nuovo Cim. A 11, 824-864.
6. Barbieri, R., Mignaco, J. A. & Remiddi, E. 1972b Electron form factors up to fourth order. II. Nuovo Cim. A 11, 865-916.
7. Erdélyi, A. (ed.) 1953 Higher transcendental functions, vol. 1. McGraw-Hill.
8. Euler, L. 1768 Institutionum calculi integralis, vol. 1, pp. 124-126. Petropoli: Impensis Academiae Imperialis Scientiarum.
9. Gröbner, W. & Hofreiter, N. 1975 Integraltafel-Zweiter Teil-Bestimmte Integrale, 5th edn. Springer.
10. Hill, C. J. 1828 Ueber die Integration logarithmisch-rationaler Differentiale. J. Reine Angew. Math. 3, 101-159.
11. αx tum quoad amplitudinem, tum quoad modulum comparandi modum exhibentis, p. 9. Lund: Academia Carolina (Lund University).
12. Hölder, O. 1928 Über eine von Abel untersuchte Transzendente und eine merkwürdige Funktionalbeziehung. Ber. über Verhandlungen Sächsischen Akad. Wissenschaften Leipzig, Math. Phys. Klasse 80, 312-325.
13. Jonquière, A. 1888 Ueber eine Klasse von Transcendenten, welche durch mehrmahlige Integration rationaler Funktionen entstehen. Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar 45, 522-531.
14. s. Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar 46, 257-268.
15. Jonquière, A. 1889b Ueber einige Transcendente, welche bei der wiederholten Integration rationaler Funktionen auftreten. Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar 15, 1-50.
16. s. Bull. Soc. Math. France 17, 142-152.
17. Kirillov, A. N. 1995 Dilogarithm identities. Prog. Theor. Phys. 118, 61-142.
18. Kölbig, K. S. 1986 Nielsen's generalized polylogarithms. SIAM J. Math. Analysis 17, 1232-1258.
19. Kölbig, K. S., Mignaco, J. A. & Remiddi, E. 1970 On Nielsen's generalized polylogarithms and their numerical calculation. BIT 10, 38-74.
20. Landen, J. 1760 A new method of computing sums of certain series. Phil. Trans. R. Soc. Lond. 51, 553-565.
21. Landen, J. 1780 Mathematical memoirs respecting a variety of subjects; with an appendix containing tables of theorems for the calculation of fluents, vol. 1. London: J. Nourse.
22. Lee, M. H. 1995 Polylogarithmic analysis of chemical potential and fluctuations in a D-dimensional free Fermi gas at low temperatures. J. Math. Phys. 36, 1217-1231.
23. Lee, M. H. 1997 Polylogarithms and Riemann's ζ function. Phys. Rev. E 56, 3909-3912.
24. Leibniz, G. W. 1855 Leibnizens mathematische Schriften (ed. C. I. Gerhardt), vol. III. Halle: H. W. Schmidt.
25. s. Acta Math. 11, 19-24.
26. Lewin, L. 1981 Polylogarithms and associated functions. Elsevier.
27. Lewin, L. (ed.) 1991 Structural properties of polylogarithms, vol. 37. Providence, RI: American Mathematical Society.
28. Maximon, L. C. & Tjon, J. A. 2000 Radiative corrections to electron-proton scattering. Phys. Rev. C 62, 054320.
29. Mignaco, J. A. & Remiddi, E. 1969 Fourth-order vacuum polarization contribution to the sixth-order electron magnetic moment. Nuovo Cim. A 60, 519-529.
30. Mo, L. W. & Tsai, Y. S. 1969 Radiative corrections to elastic and inelastic ep and μp scattering. Rev. Mod. Phys. 41, 205-235.
31. Nielsen, N. 1909 Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Abh. d. Kaiserl. Leap.-Carol. Deutschen Akademie d. Naturf. 90, 123-211.
32. Oesterlé, J. 1993 Polylogarithmes. Séminaire Bourbaki Astérisque 216, 49-67.
33. - in the Weinberg model. Nucl. Phys. B 160, 151-207.
34. Roskies, R. Z., Remiddi, E. & Levine, M. J. 1990 Quantum electrodynamics. In Advanced series on directions in high energy physics (ed. T. Kinoshita), vol. 7, pp. 162-217. World Scientific.
35. Sanchis-Lozano, M. A. 1997 Simple connections between generalized hypergeometric series and dilogarithms. J. Comput. Appl. Math. 85, 325-331.
36. Spence, W. 1809 An essay of the theory of the various orders of logarithmic transcendents; with an inquiry into their applications to the integral calculus and the summation of series. London: J. Murray.
37. Spence, W. 1820 Mathematical essays by the late William Spence, Esq. (ed. J. F. W. Herschel). Edinburgh: Oliver and Boyd.
38. 't Hooft, G. & Veltman, M. 1979 Scalar one-loop integrals. Nucl. Phys. B 153, 365-401.
39. Zagier, D. 1989 The dilogarithm function in geometry and number theory. In Number theory and related topics (papers presented at the Ramanujan Colloquium, Bombay 1988). Oxford University Press.