relation to elliptic integrals

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1: 19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

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2: 19.25 Relations to Other Functions

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§19.25(i) Legendre’s Integrals as Symmetric Integrals

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§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

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§19.25(iii) Symmetric Integrals as Legendre’s Integrals

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§19.25(vii) Hypergeometric Function

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3: 22.16 Related Functions

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Relation to Elliptic Integrals

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22.16.28

\mathcal{E}\left(x+K,k\right)=\mathcal{E}\left(x,k\right)+E\left(k\right)-k^{2% }\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right),

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Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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Relation to the Elliptic Integral $E\left(\phi,k\right)$

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Definition

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4: 14.5 Special Values

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§14.5(v) $\mu=0$ , $\nu=\pm\frac{1}{2}$

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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

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7: 19.6 Special Cases

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19.6.8

F\left(\phi,1\right)=(\sin\phi)R_{C}\left(1,{\cos}^{2}\phi\right)={% \operatorname{gd}^{-1}}\left(\phi\right).

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, ${\operatorname{gd}^{-1}}\left(\NVar{x}\right)$ : inverse Gudermannian function, $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Referenced by:: §19.10(ii), §19.10(ii)
Permalink:: http://dlmf.nist.gov/19.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.6(ii), §19.6 and Ch.19

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8: 22.15 Inverse Functions

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§22.15(ii) Representations as Elliptic Integrals

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9: 31.7 Relations to Other Functions

… ►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities

\zeta=K

K+i{K^{\prime}}

, and

i{K^{\prime}}

, where

K

and

{K^{\prime}}

are related to

k

as in §19.2(ii).

10: 20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

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