# Definite integrals of generalized certain class of incomplete elliptic integrals

## Main Article Content

## Abstract

## Article Details

*Tamkang Journal of Mathematics*,

*44*(2), 197–208. https://doi.org/10.5556/j.tkjm.44.2013.1410

## References

M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., 1964.

G. Barton, Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? III: High energies in quantum mechanics, Proc. Roy. Soc. London Ser. A, 388A (1983), 445--456.

P. J. Bushell, On a generalization of Barton's integral and related integrals of complete elliptic integrals, Math. Proc. Cambridge Philos. Soc. 101 (1987), 1--5.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Second Edition (Revised), Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band, Vol. 67, Springer-Verlag, New York, Heidelberg and Berlin, 1971.

B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, San Francisco and London, 1977.

D. Cvijovic and J. Klinowski, Integrals involving complete elliptic integrals, J. Comput. Appl. Math., 106 (1999), 169-175.

J. Das, A generalization of elliptic integrals, Bull. Calcutta Math. Soc. (Festschrift for M. Dutta), (1987), 73--82.

L. F. Epstein and J. H. Hubbell, Evaluation of a generalized elliptic- integral, J. Res. Nat. Bur. Standards B: Math. and Math. Phys., 67B (1963), 1--17.

A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vols. I and II, McGraw-Hill Book Company, New York, Toronto and London, 1953.

A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill Book Company, New York, Toronto and London, 1954.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition prepared by A. Jeffrey (and incorporating the Fourth Edition prepared by Yu. V. Geronimus and M. Yu. Tseytlin), Academic Press, New York, London, Toronto and Tokyo, 1980. 14 Shy-Der Lin, Li-Fen Chang and H. M. Srivastava.

N. T. Hai, O. I. Marichev and H. M. Srivastava, A note on the convergence of certain families of multiple hypergeometric series, J. Math. Anal. Appl., 164 (1992), 104--115.

E. L. Kaplan, Multiple elliptic integrals, J. Math. and Phys., 29 (1950), 69--75.

D. Karp, A. Savenkova and S. M. Sitnika, Series expansions for the third incomplete elliptic integral via partial fraction decompositions, Journal of Computational and Applied Mathematics, 207 (2007), 331-337.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.

S. D. Lin, Li-Fen Chang and H. M. Srivastava, A certain class of incomplete elliptic integrals and associated definite integrals, Appl. Math. Comput. (2009), doi:

1016/j.amc.2009.06.059.

S. D. Lin, S. T. Tu, H. M. Srivastava and P. Y. Wang, Some families of multiple infinite sums and associated fractional differintegral formulas for power and composite functions, J. Fract. Calc., 30 (2006), 45--58.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley and Sons, New York, Chichester, Brisbane, Toronto and Singapore, 1993.

K. F. Muller, Berechnung der Induktiviat Spulen, Arch. Elektrotech., 17 (1926), 336--353.

K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York and London, 1974.

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, Vol. 198, Academic Press, New York, London, Tokyo and Toronto, 1999.

A. P. Prudnikov, Yu. A. Bryv ckov and O. I. Mariv cev, Integrals and Series, Vol. 2: Special Functions, ''Nauka'', Moscow, 1986 (in Russian); Translated from the Russian by N. M. Queen, Second edition, Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo and Melbourne, 1988.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications ("Nauka i Tekhnika,'' Minsk, 1987), Gordon and Breach Science Publishers, Reading, Tokyo, Paris, Berlin and Langhorne (Pennsylvania), 1993.

I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill Book Company, New York, London, Sydney and Toronto, 1972.

H. M. Srivastava and S. Bromberg, Some families of generalized elliptic- integrals, Math. Comput. Modelling, 21(3) (1995), 29--38.

H. M. Srivastava and M. C. Daoust, Certain generalized Neumann expansions associated with the Kampe de Feriet function, Nederl. Akad. Wetensch. Indag = Math., 31 (1969), 449--457.

H. M. Srivastava and M. C. Daoust, A note on the convergence of Kampe de Feriet's double hypergeometric series, Math. Nachr., 53 (1972), 151--157.

H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.

H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.

H. M. Srivastava and R. N. Siddiqi, A unified presentation of certain families of elliptic- integrals related to radiation field problems, Radiat. Phys. Chem., 46 (1995), 303--315.

H. M. Srivastava, Some elliptic integrals of Barton and Bushell, J. Phys. A: Math. Gen., 28 (1995), 2305--2312.

N. A. Virchenko, On some generalizations of the functions of hypergeometric, Fract. Calc. Appl. Anal., 2(3) (1999), 233--244.

N. S. Virchenko, L. Kalla and A. Al-Zamel, Some results on a generalized hypergeometric function, Integral Transform. Spec. Funct., 12(1) (2001), 89--100.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth edition (Reprinted), Cambridge University Press, Cambridge, London and New York, 1973.