The -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.

**[1]**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.**[2]**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).**[3]**J. N. McDuff and J. R. Curreri,*Vibration control*, McGraw-Hill Book Company, Inc., New York, 1958, p. 72.**[4]**H. Kauderer,*Nichtlineare Mechanik*, Springer-Verlag, Berlin/Göttingen/Heidelberg, 1958, p. 209. MR**0145709****[5]**L. D. Landau and E. M. Lifshitz,*Mechanics*, Pergamon Press and Addison-Wesley Publishing Company, Inc., 1960, p. 27. MR**0120782****[6]**K. Pearson,*Tables of the incomplete Beta-function*, Biometrika, Cambridge University Press, 1934. MR**0226815****[7]**E. T. Whittacker and G. N. Watson,*A course of modern analysis*, Cambridge University Press, 4th edition, 1927, p. 256.**[8]**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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DOI:
https://doi.org/10.1090/qam/143948

Article copyright:
© Copyright 1963
American Mathematical Society