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Analytic structure of Schläfli function

  • Kazuhiko Aomoto (a1)

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In this note it is shown that Schläfli function can be simply expressed in terms of hyperlogarithmic functions, namely iterated integrals of forms with logarithmic poles in the sense of K. T. Chen (Theorem 1). It is also discussed the relation between Schläfli function and hypergeometric ones of Mellin-Sato type (Theorem 2). From a combinatorial point of view the structure of hyperlogarithmic functions seem very interesting just as the dilog log (so-called Abel-Rogers function) has played a crucial part in Gelfand-Gabriev-Losik’s formula of 1st Pontrjagin classes. See also [3].

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References

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[1] Aomoto, K., Les equations aux differences lineaires et les integrates des fonctions multiformes, J. Fac. Sci. Univ. Tokyo, 22 (1975), 271297.
[2] Aomoto, K., Une remarque sur la solution des equations de Schlesinger et Lapp-Danilevski II (to be submitted to J. Fac. Sci. Univ. Tokyo).
[3] Cheeger, J. and Simons, J., Differential characters and geometric invariants (preprint).
[4] Chen, K. T., Iterated integrals of differential forms and loop space homology, Ann. of Math. 97 (1973), 217246.
[5] Chen, K. T., Iterated integrals, Fundamental groups and Covering Spaces, Trans. Amer. Math. Soc. 206 (1975), 8398.
[6] Coxeter, H. S. M., The functions of Schläfli and Lobatschefsky, Quart. J. Math. 6 (1935), 1329.
[7] Deligne, P., Theorie de Hodge II, Publ. Math. I.H.E.S. 40 (1971), 558.
[8] Hopf, H., Die curvatura integra Clifford-Kleinseher Raumformen, Nach. v.d. Ges. d. Wiss., Gottingen, Mathem-physik. K.L. 1925.
[9] Iitaka, S., Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo 23 (1976), 525544.
[10] Kneser, H., Der Simplexinhalt in der nichteuklidischen Geometrie, Deutsch. Math. 1 (1936), 337340.
[11] Poincare, H., Sur la generalisation d’un theoreme elementaire de geometrie, Comptes rendus T. 140 (1905), 113117.
[12] Maier, W. und Effenberger, A., Additive Inhaltmasse im positiv gekrummten Raum, Aeq. Math. 2 (1968), 304318.
[13] Parsin, A. N., A generalization of Jacobian variety, Izv. Akad. Nauk. SSSR Ser. Mat. 30 (1966), 175182.
[14] Sato, M., Singular orbits in prehomogeneous vector spaces, Lee. Notes at Univ. of Tokyo, 1972.
[15] Schläfli, L., On the multiple integral ∫∫…∫dxdydz whose limits are p1 = a1z + b1y + … + hz > 0, p2 > 0, …, pn > 0, and x2 + y 2 + … + z 2 < 1, Quart. J. of Math. 3 (1860), 5468, 97108.
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Analytic structure of Schläfli function

  • Kazuhiko Aomoto (a1)

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