elliptic integrals

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21: 29.4 Graphics

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Figure 29.4.13: ${\mathrm{𝐸𝑐}}_{1.5}^m(x,0.5)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.85407\mathrm{\dots }$. Magnify

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Figure 29.4.14: ${\mathrm{𝐸𝑠}}_{1.5}^m(x,0.5)$ for $-2K\le x\le 2K$, $m=1,2,3$. $K=1.85407\mathrm{\dots }$. Magnify

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Figure 29.4.15: ${\mathrm{𝐸𝑐}}_{1.5}^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.4.16: ${\mathrm{𝐸𝑠}}_{1.5}^m(x,0.1)$ for $-2K\le x\le 2K$, $m=1,2,3$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.4.17: ${\mathrm{𝐸𝑐}}_{1.5}^m(x,0.9)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=2.57809\mathrm{\dots }$. Magnify

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22: 29.16 Asymptotic Expansions

… ►The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. …

23: 19.1 Special Notation

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$l,m,n$	nonnegative integers.
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$K\left(k\right),$

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$E\left(k\right),$

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$F(\varphi ,k),$

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$E(\varphi ,k),$

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24: 19.39 Software

… ►Unless otherwise stated, the functions are $K\left(k\right)$ and $E\left(k\right)$, with $0\le k^2(=m)\le 1$. … ►For other software, sometimes with $\mathrm{\Pi }({\alpha }^2,k)$ and complex variables, see the Software Index. … ►Unless otherwise stated, the variables are real, and the functions are $F(\varphi ,k)$ and $E(\varphi ,k)$. ►For research software see Bulirsch (1965b, function $\mathrm{el2}$), Bulirsch (1969b, function $\mathrm{el3}$), Jefferson (1961), and Neuman (1969a, functions $E(\varphi ,k)$ and $\mathrm{\Pi }(\varphi ,k^2,k)$). For other software, sometimes with $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ and complex variables, see the Software Index. …

25: 19.9 Inequalities

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19.9.5

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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, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i), §19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

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19.9.8

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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, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

►Further inequalities for $K\left(k\right)$ and $E\left(k\right)$ can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►Sharper inequalities for $F(\varphi ,k)$ are: … ►Inequalities for both $F(\varphi ,k)$ and $E(\varphi ,k)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). …

26: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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22.12.1 $$\tau =\mathrm{i}{K}^{\prime }\left(k\right)/K\left(k\right),$$

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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

Permalink:

http://dlmf.nist.gov/22.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.2 $$2Kk\mathrm{sn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-(n+\frac{1}{2})\tau }\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.8 $$2K\mathrm{dc}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t+\frac{1}{2}-m-n\tau }\right),$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.11 $$2K\mathrm{ns}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-n\tau }\right),$$

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Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.12 $$2K\mathrm{ds}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^{m+n}}{t-m-n\tau }\right),$$

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Symbols:

$\mathrm{ds}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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27: 22.5 Special Values

… ►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $\mathrm{i}{K}^{\prime }$. … ►

Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.

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	$z$
	$0$	$K$	$K+\mathrm{i}{K}^{\prime }$	$\mathrm{i}{K}^{\prime }$	$2K$	$2K+2\mathrm{i}{K}^{\prime }$	$2\mathrm{i}{K}^{\prime }$
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Table 22.5.2: Other special values of Jacobian elliptic functions.

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	$z$
	$\frac{1}{2}K$	$\frac{1}{2}(K+\mathrm{i}{K}^{\prime })$	$\frac{1}{2}\mathrm{i}{K}^{\prime }$
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	$\frac{3}{2}K$	$\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$	$\frac{3}{2}\mathrm{i}{K}^{\prime }$
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… ►Expansions for $K,K^{\prime }$ as $k\to 0$ or $1$ are given in §§19.5, 19.12. …

28: 19.37 Tables

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

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

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

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Functions $F(\varphi ,k)$ and $E(\varphi ,k)$

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29: 22.1 Special Notation

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$x,y$	real variables.
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$K$, $K^{\prime }$	$K\left(k\right)$, $K^{\prime }\left(k\right)=K\left(k^{\prime }\right)$ (complete elliptic integrals of the first kind (§19.2(ii))).
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$\tau$	$\mathrm{i}{K}^{\prime }/K$.

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30: 19.5 Maclaurin and Related Expansions

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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

Encodings:

pMML, png, TeX

See also:

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). … ►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product … ►Series expansions of $F(\varphi ,k)$ and $E(\varphi ,k)$ are surveyed and improved in Van de Vel (1969), and the case of $F(\varphi ,k)$ is summarized in Gautschi (1975, §1.3.2). …