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19 Elliptic IntegralsLegendre’s Integrals

§19.13 Integrals of Elliptic Integrals

Contents

§19.13(i) Integration with Respect to the Modulus

For definite and indefinite integrals of complete elliptic integrals see Byrd and Friedman (1971, pp. 610–612, 615), Prudnikov et al. (1990, §§1.11, 2.16), Glasser (1976), Bushell (1987), and Cvijović and Klinowski (1999).

For definite and indefinite integrals of incomplete elliptic integrals see Byrd and Friedman (1971, pp. 613, 616), Prudnikov et al. (1990, §§1.10.2, 2.15.2), and Cvijović and Klinowski (1994).

§19.13(ii) Integration with Respect to the Amplitude

Various integrals are listed by Byrd and Friedman (1971, p. 630) and Prudnikov et al. (1990, §§1.10.1, 2.15.1). Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for F(ϕ,k) and E(ϕ,k), together with special cases.

§19.13(iii) Laplace Transforms

For direct and inverse Laplace transforms for the complete elliptic integrals K(k), E(k), and D(k) see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.