Legendre elliptic integrals

SearchAdvancedHelp

(0.017 seconds)

21—30 of 76 matching pages

21: 19.38 Approximations

§19.38 Approximations

…

22: 29.8 Integral Equations

… ►

29.8.2 $$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{𝖯}_{\nu }\left(x\right)w(z)dz,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $w(z)$: solution and $\mu$

Referenced by:

§29.8

Permalink:

http://dlmf.nist.gov/29.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

29.8.5 $${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z_1,k^2)\frac{w_2(K)-w_2(-K)}{{d{w}_2(z)/dz|}_{z=0}}={\int }_{-K}^K{𝖯}_{\nu }\left(y\right){\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)dz,$$

ⓘ

Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

29.8.7 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z_1,k^2)\frac{w_2(K)+w_2(-K)}{w_2(0)}=-k^2\mathrm{sn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)dz,$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

►

29.8.8 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}+{d{w}_2(z)/dz|}_{z=-K}}{{d{w}_2(z)/dz|}_{z=0}}=\frac{k^2}{k^{\prime }}\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{cn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)dz,$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

29.8.9 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}-{d{w}_2(z)/dz|}_{z=-K}}{w_2(0)}=-\frac{k^4}{k^{\prime }}\mathrm{sn}(z_1,k)\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\mathrm{cn}(z,k)\frac{d^2{𝖯}_{\nu }\left(y\right)}{{dy}^2}{\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)dz.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

…

23: 19.8 Quadratic Transformations

§19.8 Quadratic Transformations

… ►showing that the convergence of $c_n$ to 0 and of $a_n$ and $g_n$ to $M(a_0,g_0)$ is quadratic in each case. … ►

Descending Landen Transformation

… ►

Ascending Landen Transformation

… ►

§19.8(iii) Gauss Transformation

…

24: 22.15 Inverse Functions

… ►

22.15.5 $$-K\le \mathrm{arcsn}(x,k)\le K,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arcsn}(x,k)$: inverse Jacobian elliptic function, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

►

22.15.6 $$0\le \mathrm{arccn}(x,k)\le 2K,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arccn}(x,k)$: inverse Jacobian elliptic function, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

►

22.15.7 $$0\le \mathrm{arcdn}(x,k)\le K,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arcdn}(x,k)$: inverse Jacobian elliptic function, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

… ►

22.15.8 $$\xi ={(-1)}^m\mathrm{arcsn}(x,k)+2mK,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arcsn}(x,k)$: inverse Jacobian elliptic function, $x$: real, $k$: modulus, $\xi$: solution and $m$: integer

Permalink:

http://dlmf.nist.gov/22.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

… ►

§22.15(ii) Representations as Elliptic Integrals

…

25: 19.10 Relations to Other Functions

… ►

§19.10(i) Theta and Elliptic Functions

…

26: 14.5 Special Values

… ►

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

… ► ►

14.5.20 $${𝖯}_{\frac{1}{2}}\left(\cos \theta \right)=\frac{2}{\pi }\left(2E\left(\sin \left(\frac{1}{2}\theta \right)\right)-K\left(\sin \left(\frac{1}{2}\theta \right)\right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable

A&S Ref:

8.13.11 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

… ►

14.5.22 $${𝖰}_{\frac{1}{2}}\left(\cos \theta \right)=K\left(\cos \left(\frac{1}{2}\theta \right)\right)-2E\left(\cos \left(\frac{1}{2}\theta \right)\right),$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable

A&S Ref:

8.13.12 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

… ►

14.5.26 $${𝑸}_{\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}\cosh \xi \mathrm{sech}\left(\frac{1}{2}\xi \right)K\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right)-4{\pi }^{-1/2}\cosh \left(\frac{1}{2}\xi \right)E\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable

A&S Ref:

8.13.7 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

…

27: 22.9 Cyclic Identities

… ►

22.9.1 $$s_{m,p}^{(2)}=\mathrm{sn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $s_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

►

22.9.2 $$c_{m,p}^{(2)}=\mathrm{cn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $c_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

►

22.9.3 $$d_{m,p}^{(2)}=\mathrm{dn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

►

22.9.4 $$s_{m,p}^{(4)}=\mathrm{sn}(z+4p^{-1}(m-1)K\left(k\right),k),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $s_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

… ►

22.9.7 $$\kappa =\mathrm{dn}(2K\left(k\right)/3,k),$$

ⓘ

Defines:

$\kappa$: change of variable (locally)

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $k$: modulus

Referenced by:

§22.9(iii), §22.9(iv)

Permalink:

http://dlmf.nist.gov/22.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

…

28: 19.11 Addition Theorems

§19.11 Addition Theorems

… ►

19.11.7 $$F(\varphi ,k)=K\left(k\right)-F(\theta ,k),$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

►

19.11.8 $$E(\varphi ,k)=E\left(k\right)-E(\theta ,k)+k^2\sin \theta \sin \varphi ,$$

ⓘ

Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

… ►

§19.11(iii) Duplication Formulas

… ►

19.11.12 $$F(\psi ,k)=2F(\theta ,k),$$

ⓘ

Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

…

29: 22.2 Definitions

… ►

22.2.1 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right),$$

ⓘ

Defines:

$q$: nome

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, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function and $k$: modulus

Referenced by:

§20.9(i), §22.16(i)

Permalink:

http://dlmf.nist.gov/22.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

… ►

22.2.3 $$\zeta =\frac{\pi z}{2K\left(k\right)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/22.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

… ►

22.2.12 $$\tau =\mathrm{i}{K}^{\prime }\left(k\right)/K\left(k\right),$$

ⓘ

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, $k$: modulus and $\tau$: ratio

Referenced by:

§22.2

Permalink:

http://dlmf.nist.gov/22.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2, §22.2 and Ch.22

…

30: 29.18 Mathematical Applications

… ►

$\beta =K+\mathrm{i}{\beta }^{\prime },$

►

$0\le {\beta }^{\prime }\le 2K^{\prime },$

►

$0\le \gamma \le 4K,$

… ►

$\alpha =K+\mathrm{i}{K}^{\prime }-{\alpha }^{\prime },$ ,

►

$\beta =K+\mathrm{i}{\beta }^{\prime },$ $0\le {\beta }^{\prime }\le 2K^{\prime },0\le \gamma \le 4K$,

…