The value of an integral in Gradshteyn and Ryzhik’s table. (English) Zbl 1429.26012
Summary: In [Notices Am. Math. Soc. 57, No. 4, 476–489 (2010; Zbl 1194.26001)], V. H. Moll observed that entry 3.248.5 in the sixth edition of I. S. Gradshteyn and I. M. Ryzhik’s table of integrals [Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. 6th ed. San Diego, CA: Academic Press (2000; Zbl 0981.65001)] was incorrect, and he asked for the value of the integral. We evaluate the integral in terms of two elliptic integrals. The evaluation is standard but involved, using real and complex analysis.
MSC:
| 26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 28A99 | Classical measure theory |
| 33E05 | Elliptic functions and integrals |
Software:
DLMFReferences:
| [1] | Amdeberhan, T.; Moll, V., The integrals in Gradshteyn and Ryzhik. Part 14: an elementary evaluation of entry 3.411.5, Sci. Ser. A Math. Sci. (N.S.), 19, 97-103 (2010) · Zbl 1244.33022 |
| [2] | Bailey, D.H., Borwein, J.M., Calkin, N.J., Girgensohn, R., Luke, D.R., Moll, V.: Experimental Mathematics in Action. A K Peters Ltd, Wellesley, MA (2007) · Zbl 1127.00002 · doi:10.1201/9781439864333 |
| [3] | Boros, G.; Moll, V.; Riley, S., An elementary evaluation of a quartic integral, Sci. Ser. A Math. Sci. (N.S.), 11, 1-12 (2005) · Zbl 1105.33001 |
| [4] | Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientist, 2nd edn. Springer, Berlin (1971). Revised · Zbl 0213.16602 · doi:10.1007/978-3-642-65138-0 |
| [5] | Gradshteyn, I.S.: Table of Integrals, Series, and Products. Edited by Jeffrey, A., Zwillinger, D., 6th edn. Academic Press, New York (2000) · Zbl 0981.65001 |
| [6] | Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Edited by Jeffrey, A., Zwillinger, D., 7th edn, Academic Press, New York (2007) · Zbl 1208.65001 |
| [7] | Moll, V., Seized opportunities, Not. Am. Math. Soc., 57, 476-484 (2010) · Zbl 1194.26001 |
| [8] | Moll, V.: Special Integrals of Gradshteyn and Ryzhik: The Proofs, vol. 1. CRC Press, Boca Raton, FA (2015) · Zbl 1347.26001 · doi:10.1201/b19419 |
| [9] | Moll, V.: Special Integrals of Gradshteyn and Ryzhik: The Proofs, vol. II. CRC Press, Boca Raton, FA (2016) · Zbl 1347.26001 |
| [10] | NIST Digital Library of Mathematical Functions. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds.). http://dlmf.nist.gov/ |
| [11] | Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis, 4th edn. Cambridge University Press, New York (1965) · JFM 45.0433.02 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
