relation to Legendre 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: 22.16 Related Functions

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Relation to Elliptic Integrals

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22.16.28 $$ℰ(x+K,k)=ℰ(x,k)+E\left(k\right)-k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.16.E28

Encodings:

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See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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Relation to the Elliptic Integral $E(\varphi ,k)$

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Definition

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3: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: 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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4: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

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5: 14.5 Special Values

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

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6: 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(iii) Symmetric Integrals as Legendre’s Integrals

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7: 22.15 Inverse Functions

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§22.15(ii) Representations as Elliptic Integrals

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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 $\pi$ 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

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Legendre’s Relation

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

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