elliptic integrals

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41: 23.7 Quarter Periods

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23.7.1 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_1\right)=e_1+\sqrt{(e_1-e_3)(e_1-e_2)}=e_1+{\omega }_1^{-2}{(K\left(k\right))}^2{k}^{\prime },$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E1

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Annotations for §23.7 and Ch.23

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23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E2

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Annotations for §23.7 and Ch.23

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23.7.3 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_3\right)=e_3-\sqrt{(e_1-e_3)(e_2-e_3)}=e_3-{\omega }_1^{-2}{(K\left(k\right))}^{2}k,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E3

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Annotations for §23.7 and Ch.23

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42: 19.12 Asymptotic Approximations

§19.12 Asymptotic Approximations

►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: ►

19.12.1 $$K\left(k\right)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}k_{}^{\prime }{}_{}{}^{2m}\left(\ln \left(\frac{1}{k^{\prime }}\right)+d(m)\right),$$ ,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $d(m)$: function

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E1

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Annotations for §19.12 and Ch.19

… ►For the asymptotic behavior of $F(\varphi ,k)$ and $E(\varphi ,k)$ as $\varphi \to \frac{1}{2}\pi -$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►Asymptotic approximations for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$, with different variables, are given in Karp et al. (2007). …

43: 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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44: 23.4 Graphics

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Figure 23.4.7: $\mathrm{\wp }\left(x\right)$ with ${\omega }_1=K\left(k\right)$, ${\omega }_3=\mathrm{i}{K}^{\prime }\left(k\right)$ for $0\le x\le 9$, $k^2$ = 0. … Magnify

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Figure 23.4.8: $\mathrm{\wp }\left(x+\mathrm{i}y\right)$ with ${\omega }_1=K\left(k\right)$, ${\omega }_3=\mathrm{i}{K}^{\prime }\left(k\right)$ for $-2K\left(k\right)\le x\le 2K\left(k\right)$, $0\le y\le 6K^{\prime }\left(k\right)$, $k^2=0.9$. … Magnify 3D Help

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45: 22.2 Definitions

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

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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

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Annotations for §22.2 and Ch.22

►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ►

$K\left(k\right)={\displaystyle \frac{\pi }{2}}{\theta }_3^2(0,q),$

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

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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:

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Annotations for §22.2 and Ch.22

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

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

Referenced by:

§22.2

Permalink:

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

Encodings:

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See also:

Annotations for §22.2, §22.2 and Ch.22

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46: 22.20 Methods of Computation

… ►This formula for $\mathrm{dn}$ becomes unstable near $x=K$. If only the value of $\mathrm{dn}(x,k)$ at $x=K$ is required then the exact value is in the table 22.5.1. … ►If either $\tau$ or $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ is given, then we use $k={\theta }_2^2(0,q)/{\theta }_3^2(0,q)$, $k^{\prime }={\theta }_4^2(0,q)/{\theta }_3^2(0,q)$, $K=\frac{1}{2}\pi {\theta }_3^2(0,q)$, and $K^{\prime }=-\mathrm{i}\tau K$, obtaining the values of the theta functions as in §20.14. ►If $k,k^{\prime }$ are given with $k^2+k_{}^{\prime }{}_{}{}^2=1$ and , then $K,K^{\prime }$ can be found from … ►

$K^{\prime }={\displaystyle \frac{\pi }{2M(1,k)}},$

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47: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

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19.14.1 $${\int }_1^x\frac{dt}{\sqrt{t^3-1}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}+1-x}{\sqrt{3}-1+x}}$, $k^2={\displaystyle \frac{2-\sqrt{3}}{4}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i)

Permalink:

http://dlmf.nist.gov/19.14.E1

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Annotations for §19.14(i), §19.14 and Ch.19

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19.14.2 $${\int }_x^1\frac{dt}{\sqrt{1-t^3}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}-1+x}{\sqrt{3}+1-x}}$, $k^2={\displaystyle \frac{2+\sqrt{3}}{4}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.14.E2

Encodings:

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Annotations for §19.14(i), §19.14 and Ch.19

… ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. …

48: 22.15 Inverse Functions

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

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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

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

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

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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

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

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

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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:

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

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§22.15(ii) Representations as Elliptic Integrals

… ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …

49: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $F_D$ (Carlson (1961b)). … ► … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

50: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►

Table 29.3.1: Eigenvalues of Lamé’s equation.

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eigenvalue $h$	parity	period
$a_{\nu }^{2m}\left(k^2\right)$	even	$2K$
$a_{\nu }^{2m+1}\left(k^2\right)$	odd	$4K$
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… ►In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$. … ►To complete the definitions, ${\mathrm{𝐸𝑐}}_{\nu }^m(K,k^2)$ is positive and ${d{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)/dz|}_{z=K}$ is negative. …