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The dilogarithm in algebraic fields

Published online by Cambridge University Press:  09 April 2009

L. Lewin
L. Lewin
Affiliation:
980 McIntireBoulder, Colorado 80303, U.S.A.
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Abstract

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Special numerical formulae for the dilogarithm of powers of a base quantity have been shown to exist by Watson, Coxeter and others. Abel's equation for the dilogarithm was put in this “exponent” form, as a result of which a four-variable, symmetrical equation, also in exponent form, was deduced. From these equations a large number of special numerical results were produced, from which certain properties of a general structural nature emerged, and enabled two new results to be predicted.

The algebraic bases for these results can be grouped in trigonometric or non-trigonometric form, and for the former it seems to be necessary to examine the properties of the dilogarithm in the complex plane. Even so, there are some identities that seem to be outside the scope of the present methods.

It is speculated that certain factorization relations, which can be identified from the equations, may play a substantial role in the results; but so far no analytic derivation of this property has appeared possible.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 3 , December 1982 , pp. 302 - 330
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Abel, N. H., Oeuvres complètes, Vol. 2, p. 193.Google Scholar
[2]Chudnovsky, G. V., ‘Padé approximations to the generalized hypergeometric functions’, J. Math. Pures Appl. 58 (4) (1979), 445–476.Google Scholar
[3]Clausen, T., J Reine Angew. Math. 8 (1832), 298–300.Google Scholar
[4]Coxeter, H. S. M., ‘The functions of Schläfli and Lobatschefsky’, Quart J. Math. Oxford Ser. 6 (1935), 13–29.CrossRefGoogle Scholar
[5]Hill, C. J., Specimen exercitii analytici etc., Lund, (1830) p. 9.Google Scholar
[6]Kummer, E. E., J. Reine Angew. Math. 21 (1840), 74–90.Google Scholar
[7]Landen, J., Mathematical memoirs 1 pp. 112–118 (1780).Google Scholar
[8]Legendre, A. M., Exercises de calcul intégral, Vol. 1 (1811), p. 247.Google Scholar
[9]Lewin, L., Polylogarithms and associated functions (Elsevier — North-Holland, New York, Oxford, 1981).Google Scholar
[10]Richmond, B. and Szekeres, G., ‘Some formulas related to dilogarithms, the zeta function and the Andrews-Gordon identities’, J. Austral. Math. Soc. 31 (1981), 362–373.CrossRefGoogle Scholar
[11]Rogers, L. J., ‘On function sum theorems connected with the series , Proc. London Math. Soc. 4 (1907), 169–189.CrossRefGoogle Scholar
[12]Schaeffer, W., J. Reine Angew. Math. 30 (1846), 288.Google Scholar
[13]Spence, W., An essay on logarithmic transcendents (London and Edinburgh, 1809).Google Scholar
[14]Watson, G. N., ‘A note on Spence's logarithmic transcendent’, Quart J. Math. Oxford Ser. 8 (1937), 39–42.CrossRefGoogle Scholar