Inequalities for a symmetric elliptic integral
Author:
B. C. Carlson
Journal:
Proc. Amer. Math. Soc. 25 (1970), 698-703
MSC:
Primary 33.19
DOI:
https://doi.org/10.1090/S0002-9939-1970-0257412-X
MathSciNet review:
0257412
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Abstract | References | Similar Articles | Additional Information
Abstract: Inequalities are found for an incomplete elliptic integral of the first kind which represents the reciprocal of the capacity of an ellipsoid with semiaxes $x,\;y,\;z$. One sequence of symmetric algebraic functions of $x,\;y,\;z$ converges to the value of the integral from below and two from above. Among the elements of these sequences are upper and lower approximations due to Pólya and Szegö.
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- B. C. Carlson, Normal elliptic integrals of the first and second kinds, Duke Math. J. 31 (1964), 405–419. MR 164067 G. Pólya, Arch. Math. und Physik (3) 26 (1917), 65. G. Szegö, Arch. Math. und Physik (3) 28 (1920), 81-82. H. Poincaré, Sur un théorème de M. Liapounoff, relalif à l’équilibre d’une masse fluide, C. R. Acad. Sci. Paris 104 (1887), 622-625.
- D. G. Zill and B. C. Carlson, Symmetric elliptic integrals of the third kind, Math. Comp. 24 (1970), 199–214. MR 262553, DOI https://doi.org/10.1090/S0025-5718-1970-0262553-5 B. C. Carlson, Inequality of mixed arithmetic and geometric means, SIAM Rev. (to appear).
- B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 17 (1966), 32–39. MR 188497, DOI https://doi.org/10.1090/S0002-9939-1966-0188497-6
- B. C. Carlson, Some series and bounds for incomplete elliptic integrals, J. Math. and Phys. 40 (1961), 125–134. MR 130398
- B. C. Carlson, Hidden symmetries of special functions, SIAM Rev. 12 (1970), 332–345. MR 262554, DOI https://doi.org/10.1137/1012078 G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952. MR 13, 727.
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Additional Information
Keywords:
Elliptic integrals,
inequalities,
ellipsoid,
capacity,
duplication theorem,
hypergeometric <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$R$">-functions
Article copyright:
© Copyright 1970
American Mathematical Society