elliptic integrals

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1: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … … ►

§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

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2: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

… ►is called an elliptic integral. … ►

§19.2(ii) Legendre’s Integrals

… ►The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. … ►

§19.2(iii) Bulirsch’s Integrals

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3: 36.2 Catastrophes and Canonical Integrals

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Canonical Integrals

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${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathrm{𝟎}\right)=\frac{1}{3}\sqrt{\pi }\mathrm{\Gamma }\left(\frac{1}{6}\right)=3.28868,$

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36.2.25 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(x,-y,z\right)={\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right).$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

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36.2.26 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(-\frac{1}{2}x\mp \frac{\sqrt{3}}{2}y,\pm \frac{\sqrt{3}}{2}x-\frac{1}{2}y,z\right)={\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right),$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

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36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E28

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

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4: 29.10 Lamé Functions with Imaginary Periods

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29.10.2 $$z^{\prime }=\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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${\mathrm{𝐸𝑐}}_{\nu }^{2m}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

… ►The first and the fourth functions have period $2\mathrm{i}{K}^{\prime }$; the second and the third have period $4\mathrm{i}{K}^{\prime }$. …

5: 29.13 Graphics

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Figure 29.13.5: ${\mathrm{𝑢𝐸}}_4^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.13.6: ${\mathrm{𝑢𝐸}}_4^m(x,0.9)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=2.57809\mathrm{\dots }$. Magnify

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Figure 29.13.21: $|{\mathrm{𝑢𝐸}}_4^1(x+\mathrm{i}y,0.1)|$ for $-3K\le x\le 3K$, $0\le y\le 2K^{\prime }$. $K=1.61244\mathrm{\dots }$, $K^{\prime }=2.57809\mathrm{\dots }$. Magnify 3D Help

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Figure 29.13.22: $|{\mathrm{𝑢𝐸}}_4^1(x+\mathrm{i}y,0.5)|$ for $-3K\le x\le 3K$, $0\le y\le 2K^{\prime }$. $K=K^{\prime }=1.85407\mathrm{\dots }$. Magnify 3D Help

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Figure 29.13.23: $|{\mathrm{𝑢𝐸}}_4^1(x+\mathrm{i}y,0.9)|$ for $-3K\le x\le 3K$, $0\le y\le 2K^{\prime }$. $K=2.57809\mathrm{\dots }$, $K^{\prime }=1.61244\mathrm{\dots }$. Magnify 3D Help

6: 22.21 Tables

… ►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. … ►Zhang and Jin (1996, p. 678) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …

7: 19.4 Derivatives and Differential Equations

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§19.4(i) Derivatives

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${\displaystyle \frac{d(E\left(k\right)-k_{}^{\prime }{}_{}{}^{2}K\left(k\right))}{dk}}=kK\left(k\right),$

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${\displaystyle \frac{dE\left(k\right)}{dk}}={\displaystyle \frac{E\left(k\right)-K\left(k\right)}{k}},$

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${\displaystyle \frac{d(E\left(k\right)-K\left(k\right))}{dk}}=-{\displaystyle \frac{kE\left(k\right)}{k_{}^{\prime }{}_{}{}^2}},$

… ►If $\varphi =\pi /2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …

8: 19.6 Special Cases

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§19.6(i) Complete Elliptic Integrals

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$K\left(0\right)=E\left(0\right)=K^{\prime }\left(1\right)=E^{\prime }\left(1\right)=\frac{1}{2}\pi ,$

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$K\left(1\right)=K^{\prime }\left(0\right)=\mathrm{\infty },$

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§19.6(ii) $F(\varphi ,k)$

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§19.6(iii) $E(\varphi ,k)$

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9: 19.7 Connection Formulas

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19.7.1 $$E\left(k\right)K^{\prime }\left(k\right)+E^{\prime }\left(k\right)K\left(k\right)-K\left(k\right)K^{\prime }\left(k\right)=\frac{1}{2}\pi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind and $k$: real or complex modulus

Referenced by:

§19.21(i), §19.35(i), §19.7(i)

Permalink:

http://dlmf.nist.gov/19.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(i), §19.7(i), §19.7 and Ch.19

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$$K\left(1/k\right)=k(K\left(k\right)\mp \mathrm{i}K\left(k^{\prime }\right)),$$

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$$K\left(1/k^{\prime }\right)=k^{\prime }(K\left(k^{\prime }\right)\pm \mathrm{i}K\left(k\right)),$$

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$E(\varphi ,k_1)=(E(\beta ,k)-k_{}^{\prime }{}_{}{}^{2}F(\beta ,k))/k,$

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

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10: 29.14 Orthogonality

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29.14.2 $$⟨g,h⟩={\int }_0^K{\int }_0^{K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.3 $$w(s,t)={\mathrm{sn}}^2(K+\mathrm{i}t,k)-{\mathrm{sn}}^2(s,k).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.11 $$⟨g,h⟩={\int }_0^{4K}{\int }_0^{2K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Permalink:

http://dlmf.nist.gov/29.14.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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