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On a generalization of Barton’s integral and related integrals of complete elliptic integrals. (English) Zbl 0628.33002

The author evaluates certain integrals of the form

${\int }_{0}^{1}{k}^{\mu }f\left(k\right)dk,$

where f(k) is replaced by K(k), E(k), K(k’) and E(k’), the complete elliptic integrals of the first and second kind respectively, ${k}^{\text{'}}=\sqrt{\left(1-{k}^{2}\right)}$. The results can be deduced from the known integrals involving hypergeometric functions [A. Erdélyi et al.: Higher transcendental functions. II (1953; Zbl 0052.295)]. M. L. Glasser [J. Res. Natl. Bur. Stand., Sect. B 80, 313- 323 (1976; Zbl 0339.33001)] has given an extensive table of definite integrals of the complete elliptic integral K.

Generalized elliptic-type integrals have been considered by L. F. Epstein and J. H. Hubbell [ibid. 67, 1-17 (1963; Zbl 0114.064)], the reviewer, S. Conde and J. H. Hubbell [Appl. Anal. 22, 273-287 (1986; Zbl 0588.33001)], the reviewer, C. Leubner and J. H. Hubbell [Appl. Anal. 25, 269-274 (1987; Zbl 0606.33003)] and the reviewer and B. Al-Saqabi [Rev. Bras. Fisica 16, 145-156 (1986; Zbl 0597.33004)].

Reviewer: S.L.Kalla

##### MSC:
 33E05 Elliptic functions and integrals 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$
##### Keywords:
elliptic-type integrals