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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. (English) Zbl 1118.33010

The authors obtain two convergent series expansions for the incomplete elliptic integral of the first kind \[ F(\lambda,k)=\int_0^\lambda\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}} \] valid at any point in the unit square \(0<\lambda, k<1\). These expansions are expressed in terms of recursively computed elementary functions. The expansions are truncated after \(N\) terms and, by expressing the tails as integrals combined with use of bounds for certain hypergeometric functions, explicit bounds for the remainders \(R_N\) are obtained.
The truncated expansions yield asymptotic approximations for \(F(\lambda,k)\) as \(\lambda\) and/or \(k\) approach unity. The approximations also remain valid as the logarithmic singularity \(\lambda=k=1\) is approached in any direction. The first two approximations complete with error bounds are presented explicitly and numerical calculations are given to illustrate their accuracy.

MSC:

33E05 Elliptic functions and integrals
33C75 Elliptic integrals as hypergeometric functions
33F05 Numerical approximation and evaluation of special functions
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[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1970), Dover: Dover New York · Zbl 0515.33001
[2] Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists (1971), Spinger: Spinger New York · Zbl 0213.16602
[3] Carlson, B. C., Some series and bounds for incomplete elliptic integrals, J. Math. Phys., 40, 125-134 (1961) · Zbl 0113.28103
[4] Carlson, B. C., Special Functions of Applied Mathematics (1977), Academic Press: Academic Press New York · Zbl 0394.33001
[5] Carlson, B. C.; Gustafson, J. L., Asymptotic expansion of the first elliptic integral, SIAM J. Math.Anal., 16, 5, 1072-1092 (1985) · Zbl 0593.33002
[6] Carlson, B. C.; Gustafson, J. L., Asymptotic approximations for symmetric elliptic integrals, SIAM J. Math. Anal., 25, 288-303 (1994) · Zbl 0794.41021
[7] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 1, McGraw-Hill Book Company, Inc., New York, 1953.; A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 1, McGraw-Hill Book Company, Inc., New York, 1953.
[8] J.L. Gustafson, Asymptotic Formulas for Elliptic Integrals, Ph.D. Thesis, Iowa State University, Ames, IA, 1982.; J.L. Gustafson, Asymptotic Formulas for Elliptic Integrals, Ph.D. Thesis, Iowa State University, Ames, IA, 1982.
[9] D. Karp, A. Savenkova, S.M. Sitnik, Series expansions and asymptotics for the third incomplete elliptic integral via partial fraction decompositions, J. Comput. Appl. Math. 2006, to appear.; D. Karp, A. Savenkova, S.M. Sitnik, Series expansions and asymptotics for the third incomplete elliptic integral via partial fraction decompositions, J. Comput. Appl. Math. 2006, to appear. · Zbl 1117.33013
[10] Kaplan, E. L., Auxiliary table for the incomplete elliptic integrals, J. Math. Phys., 27, 11-36 (1948) · Zbl 0031.22303
[11] Kelisky, R. P., Inverse elliptic functions and Legendre polynomials, Amer. Math. Monthly, 66, 480-483 (1959) · Zbl 0087.08502
[12] Koepf, W., Hypegeometric Summation (1998), Advanced Lectures in Mathematics: Advanced Lectures in Mathematics Vieweg
[13] López, J. L., Asymptotic expansions of symmetric standard elliptic integrals, SIAM J. Math. Anal., 31, 4, 754-775 (2000) · Zbl 0994.33010
[14] López, J. L., Uniform asymptotic expansions of symmetric elliptic integrals, Constructive Approximation, 17, 4, 535-559 (2001) · Zbl 1086.41015
[15] J.L. López, Asymptotic expansions of Mellin convolutions by means of analytic continuation, J. Comp. Appl. Math. 2006, to appear.; J.L. López, Asymptotic expansions of Mellin convolutions by means of analytic continuation, J. Comp. Appl. Math. 2006, to appear.
[16] Mitrinovic, D. S.; Pecaric, J. E.; Fink, A. M., Classical and New Inequalities in Analysis (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0771.26009
[17] Nellis, W. J.; Carlson, B. C., Reduction and evaluation of elliptic integrals, Math. Comput., 20, 223-231 (1966) · Zbl 0135.38702
[18] Ponnusamy, S.; Vuorinen, M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44, 278-301 (1997) · Zbl 0897.33001
[19] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, vol. 3, More Special Functions, Gordon and Breach Science Publishers, 1990.; A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, vol. 3, More Special Functions, Gordon and Breach Science Publishers, 1990. · Zbl 0967.00503
[20] Radon, B., Sviluppi in serie degli integrali ellipttici, Atti. Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat. Ser., 2, 8, 69-109 (1950)
[21] S.M. Sitnik, Inequalities for the Legendre complete elliptic integrals, Preprint of the Far-Eastern Branch of the Russian Academy of Sciences, Vladivostok, 1994.; S.M. Sitnik, Inequalities for the Legendre complete elliptic integrals, Preprint of the Far-Eastern Branch of the Russian Academy of Sciences, Vladivostok, 1994.
[22] Sitnik, S. M., Refinements of integral Cauchy-Bunyakovskii inequality, Vestnik Samara State Technical Univ., 9, 37-45 (2000)
[23] Sitnik, S. M., Refinements of Cauchy-Bunyakovskii inequality and applications, Vestnik Samara State Acad. Econ., 1, 8, 302-313 (2002)
[24] Sitnik, S. M., Means and generalizations of Cauchy-Bunyakovskii inequalities and applications, Sci. Res. Chernozemie, 1, 3-42 (2005)
[25] G. Szegő, Orthogonal polynomials, AMS Colloquium Publications vol. 23, 1991 (8th printing).; G. Szegő, Orthogonal polynomials, AMS Colloquium Publications vol. 23, 1991 (8th printing). · JFM 65.0278.03
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