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relation to Legendre elliptic integrals

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1: 19.10 Relations to Other Functions
§19.10(i) Theta and Elliptic Functions
2: 22.16 Related Functions
Relation to Elliptic Integrals
Relation to the Elliptic Integral E ( ϕ , k )
Definition
3: 19.6 Special Cases
4: 19.5 Maclaurin and Related Expansions
§19.5 Maclaurin and Related Expansions
5: 19.25 Relations to Other Functions
§19.25(i) Legendre’s Integrals as Symmetric Integrals
§19.25(iii) Symmetric Integrals as Legendre’s Integrals
6: 14.5 Special Values
§14.5(v) μ = 0 , ν = ± 1 2
7: 22.15 Inverse Functions
§22.15(ii) Representations as Elliptic Integrals
8: 19.35 Other Applications
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute π to high precision (Borwein and Borwein (1987, p. 26)). …
9: 19.36 Methods of Computation
Because of cancellations in (19.26.21) it is advisable to compute R G from R F and R D by (19.21.10) or else to use §19.36(ii). Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). … For series expansions of Legendre’s integrals see §19.5. …
10: 19.7 Connection Formulas
§19.7 Connection Formulas
Legendre’s Relation
Reciprocal-Modulus Transformation
Imaginary-Modulus Transformation
§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k )