The $Ateb(h)$-functions and their properties
Author:
R. M. Rosenberg
Journal:
Quart. Appl. Math. 21 (1963), 37-47
MSC:
Primary 33.15
DOI:
https://doi.org/10.1090/qam/143948
MathSciNet review:
143948
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Abstract: The Ateb(h)-functions are inversions of incomplete Beta-functions. They are the solutions of normal mode vibrations of certain nonlinear multi-degree-of-freedom systems just as the trigonometric functions yield the normal mode vibrations of the corresponding linear systems. Like elliptic functions, the Ateb(h)-functions depend on a parameter. The Ateb-functions reduce to trigonometric functions, and the Atebh-functions to hyperbolic functions when the parameter is 1. When the parameter is 2, the Ateb-functions become elliptic functions. A number of properties of the Ateb(h)-functions, such as identities, derivatives, integrals, differential equations satisfied by them, etc., are given.
R. M. Rosenberg and C. S. Hsu, On the geometrization of normal vibrations of nonlinear systems having many degrees of freedom, to be published in the Proceedings, IUTAM International Symposium on Nonlinear Oscillations, Kiev, U.S.S.R., September 1961.
A. D. Miishkes, On the exactness of approximate methods of analyzing small nonlinear vibrations of one degree of freedom, in "Dynamic problems, Collected Papers," vol. 1, Riga, 1953, p. 139 (in Russian).
J. N. McDuff and J. R. Curreri, Vibration control, McGraw-Hill Book Company, Inc., New York, 1958, p. 72.
- Hans Kauderer, Nichtlineare Mechanik, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958 (German). MR 0145709
- L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics, Vol. 1, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. Translated from the Russian by J. B. Bell. MR 0120782
- Karl Pearson (ed.), Tables of the incomplete beta-function, Published for the Biometrika Trustees at the Cambridge University Press, London, 1968. Originally prepared under the direction of and edited by Karl Pearson; Second edition with a new introduction by E. S. Pearson and N. L. Johnson. MR 0226815
E. T. Whittacker and G. N. Watson, A course of modern analysis, Cambridge University Press, 4th edition, 1927, p. 256.
See for instance, T. von Karman and M. A. Biot, Mathematical methods of engineering, McGraw-Hill Book Company, Inc., New York and London, 1st edition, 1940, p. 121.
R. M. Rosenberg and C. S. Hsu, On the geometrization of normal vibrations of nonlinear systems having many degrees of freedom, to be published in the Proceedings, IUTAM International Symposium on Nonlinear Oscillations, Kiev, U.S.S.R., September 1961.
A. D. Miishkes, On the exactness of approximate methods of analyzing small nonlinear vibrations of one degree of freedom, in "Dynamic problems, Collected Papers," vol. 1, Riga, 1953, p. 139 (in Russian).
J. N. McDuff and J. R. Curreri, Vibration control, McGraw-Hill Book Company, Inc., New York, 1958, p. 72.
H. Kauderer, Nichtlineare Mechanik, Springer-Verlag, Berlin/Göttingen/Heidelberg, 1958, p. 209.
L. D. Landau and E. M. Lifshitz, Mechanics, Pergamon Press and Addison-Wesley Publishing Company, Inc., 1960, p. 27.
K. Pearson, Tables of the incomplete Beta-function, Biometrika, Cambridge University Press, 1934.
E. T. Whittacker and G. N. Watson, A course of modern analysis, Cambridge University Press, 4th edition, 1927, p. 256.
See for instance, T. von Karman and M. A. Biot, Mathematical methods of engineering, McGraw-Hill Book Company, Inc., New York and London, 1st edition, 1940, p. 121.
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Article copyright:
© Copyright 1963
American Mathematical Society
