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relation to elliptic integrals

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1: 19.10 Relations to Other Functions
§19.10(i) Theta and Elliptic Functions
2: 19.25 Relations to Other Functions
§19.25(i) Legendre’s Integrals as Symmetric Integrals
§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
§19.25(iii) Symmetric Integrals as Legendre’s Integrals
§19.25(vii) Hypergeometric Function
3: 22.16 Related Functions
Relation to Elliptic Integrals
Relation to the Elliptic Integral E ( ϕ , k )
Definition
4: 14.5 Special Values
§14.5(v) μ = 0 , ν = ± 1 2
5: 15.17 Mathematical Applications
Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …
6: 19.5 Maclaurin and Related Expansions
§19.5 Maclaurin and Related Expansions
7: 19.6 Special Cases
8: 22.15 Inverse Functions
§22.15(ii) Representations as Elliptic Integrals
9: 31.7 Relations to Other Functions
Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities ζ = K , K + i K , and i K , where K and K are related to k as in §19.2(ii).
10: 20.9 Relations to Other Functions
§20.9(i) Elliptic Integrals