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21: 19.2 Definitions

… ►The cases with $\varphi =\pi /2$ are the complete integrals: … ►The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. ►Legendre’s complementary complete elliptic integrals are defined via …For more details on the analytical continuation of these complete elliptic integrals see Lawden (1989, §§8.12–8.14). … ►The integrals are complete if $x=\mathrm{\infty }$. …

22: 19.5 Maclaurin and Related Expansions

… ►

19.5.5 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right)=r+8r^2+84r^3+992r^4+\mathrm{\cdots },$$ $r=\frac{1}{16}{k}^2$, $0\le k\le 1$.

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

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E5

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Annotations for §19.5 and Ch.19

… ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). ►

19.5.8 $$K\left(k\right)=\frac{\pi }{2}{\left(1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\right)}^2,$$ ,

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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, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E8

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Annotations for §19.5 and Ch.19

►

19.5.9 $$E\left(k\right)=K\left(k\right)+\frac{2{\pi }^2}{K\left(k\right)}\frac{{\sum }_{n=1}^{\mathrm{\infty }}{(-1)}^n{n}^2{q}^{n^2}}{1+2{\sum }_{n=1}^{\mathrm{\infty }}{(-1)}^n{q}^{n^2}},$$ .

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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, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E9

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Annotations for §19.5 and Ch.19

►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product …

23: 21.11 Software

… ►A more complete list of available software for computing these functions is found in the Software Index. …

24: 23.7 Quarter Periods

… ►

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

…

25: 29.18 Mathematical Applications

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

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

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

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

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

…

26: 19.37 Tables

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§19.37(ii) Legendre’s Complete Integrals

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Functions $K\left(k\right)$ and $E\left(k\right)$

►Tabulated for $k^2=0(.01)1$ to 6D by Byrd and Friedman (1971), to 15D for $K\left(k\right)$ and 9D for $E\left(k\right)$ by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964). … ►

Functions $K\left(k\right)$, $K^{\prime }\left(k\right)$, and $\mathrm{i}{K}^{\prime }\left(k\right)/K\left(k\right)$

… ►

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

…

27: 29.2 Differential Equations

… ►This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K}^{\prime }$, where $p,q\in \mathbb{Z}$, and $K$, $K^{\prime }$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime }(={(1-k^2)}^{1/2})$, respectively; see §19.2(ii). … ►

29.2.8 $$\eta ={(e_1-e_3)}^{-1/2}(z-\mathrm{i}{K}^{\prime }),$$

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Defines:

$\eta$ (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/29.2.E8

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Annotations for §29.2(ii), §29.2 and Ch.29

…

28: 29.8 Integral Equations

… ►Let $w(z)$ be any solution of (29.2.1) of period $4K$, $w_2(z)$ be a linearly independent solution, and $𝒲\left\{w,w_2\right\}$ denote their Wronskian. … ►

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

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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

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Annotations for §29.8 and Ch.29

… ►

$$w(z+2K)=\sigma w(z),$$

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$$w_2(z+2K)=\tau w(z)+\sigma w_2(z).$$

… ►

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

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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:

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Annotations for §29.8 and Ch.29

…

29: 15.17 Mathematical Applications

… ►

§15.17(ii) Conformal Mappings

… ►Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. …

30: 19.1 Special Notation

… ►The first set of main functions treated in this chapter are Legendre’s complete integrals …The functions (19.1.1) and (19.1.2) are used in Erdélyi et al. (1953b, Chapter 13), except that $\mathrm{\Pi }({\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ are denoted by ${\mathrm{\Pi }}_1(\nu ,k)$ and $\mathrm{\Pi }(\varphi ,\nu ,k)$, respectively, where $\nu =-{\alpha }^2$. ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). … ► $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ are the symmetric (in $x$, $y$, and $z$) integrals of the first, second, and third kinds; they are complete if exactly one of $x$, $y$, and $z$ is identically 0. … ►The first three functions are incomplete integrals of the first, second, and third kinds, and the $\mathrm{cel}$ function includes complete integrals of all three kinds.