22 Jacobian Elliptic Functions Properties 22.15 Inverse Functions 22.17 Moduli Outside the Interval [0,1]

§22.16 Related Functions

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Permalink:: http://dlmf.nist.gov/22.16
See also:: Annotations for Ch.22

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

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Keywords:: amplitude ( $\operatorname{am}$ ) function
Notes:: See Walker (1996, pp. 138–141).
Referenced by:: §19.25(v), §22.14(i), §29.1, §29.2(ii)
Permalink:: http://dlmf.nist.gov/22.16.i
See also:: Annotations for §22.16 and Ch.22

Definition

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Keywords:: amplitude ( $\operatorname{am}$ ) function, definition
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.1		$\operatorname{am}\left(x,k\right)=\operatorname{Arcsin}\left(\operatorname{sn}% \left(x,k\right)\right),$
$x\in\mathbb{R}$ ,
ⓘ Defines: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\in$ : element of, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $\mathbb{R}$ : real line, $x$ : real and $k$ : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E1 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

where the inverse sine has its principal value when $-K\leq x\leq K$ and is defined by continuity elsewhere. See Figure 22.16.1. $\operatorname{am}\left(x,k\right)$ is an infinitely differentiable function of $x$ .

Quasi-Periodicity

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Keywords:: amplitude ( $\operatorname{am}$ ) function, quasi-periodicity
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.2		$\operatorname{am}\left(x+2K,k\right)=\operatorname{am}\left(x,k\right)+\pi.$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E2 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Integral Representation

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Keywords:: amplitude ( $\operatorname{am}$ ) function, integral representation
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.3		$\operatorname{am}\left(x,k\right)=\int_{0}^{x}\operatorname{dn}\left(t,k\right% )\,\mathrm{d}t.$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus Referenced by: Figure 22.16.2, Figure 22.16.2 Permalink: http://dlmf.nist.gov/22.16.E3 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Special Values

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Keywords:: Gudermannian function, amplitude ( $\operatorname{am}$ ) function, relation to amplitude ( $\operatorname{am}$ ) function, special values
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.4	$\displaystyle\operatorname{am}\left(x,0\right)$	$\displaystyle=x,$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function and $x$ : real Permalink: http://dlmf.nist.gov/22.16.E4 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22
22.16.5	$\displaystyle\operatorname{am}\left(x,1\right)$	$\displaystyle=\operatorname{gd}\left(x\right).$
ⓘ Symbols: $\operatorname{gd}\NVar{x}$ : Gudermannian function, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function and $x$ : real Permalink: http://dlmf.nist.gov/22.16.E5 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

For the Gudermannian function $\operatorname{gd}\left(x\right)$ see §4.23(viii).

Approximation for Small $x$

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Keywords:: amplitude ( $\operatorname{am}$ ) function, approximations, small $x$
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.6		$\operatorname{am}\left(x,k\right)=x-k^{2}\frac{x^{3}}{3!}+k^{2}\left(4+k^{2}% \right)\frac{x^{5}}{5!}+O\left(x^{7}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $!$ : factorial (as in $n!$ ), $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E6 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Approximations for Small $k$ , $k^{\prime}$

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Keywords:: amplitude ( $\operatorname{am}$ ) function, approximations, relation to Gudermannian function, small $k,k^{\prime}$
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.7		$\operatorname{am}\left(x,k\right)=x-\tfrac{1}{4}k^{2}(x-\sin x\cos x)+O\left(k% ^{4}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $x$ : real and $k$ : modulus A&S Ref: 16.13.4 Permalink: http://dlmf.nist.gov/22.16.E7 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.8		$\operatorname{am}\left(x,k\right)=\operatorname{gd}x-\tfrac{1}{4}{k^{\prime}}^% {2}(x-\sinh x\cosh x)\operatorname{sech}x+O\left({k^{\prime}}^{4}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{gd}\NVar{x}$ : Gudermannian function, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.15.4 Permalink: http://dlmf.nist.gov/22.16.E8 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Fourier Series

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Keywords:: Fourier series, amplitude ( $\operatorname{am}$ ) function
See also:: Annotations for §22.16(i), §22.16 and Ch.22

With $q$ as in (22.2.1) and $\zeta=\pi x/(2K)$ ,

22.16.9		$\operatorname{am}\left(x,k\right)=\frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^% {n}\sin\left(2n\zeta\right)}{n(1+q^{2n})}.$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $q$ : nome, $\sin\NVar{z}$ : sine function, $x$ : real and $k$ : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E9 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Relation to Elliptic Integrals

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Keywords:: Legendre’s elliptic integrals, $\operatorname{am}$ function, amplitude ( $\operatorname{am}$ ) function, relation to elliptic integrals, relations to other functions
See also:: Annotations for §22.16(i), §22.16 and Ch.22

If $-K\leq x\leq K$ , then the following four equations are equivalent:

22.16.10		$x=F\left(\phi,k\right),$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E10 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.11		$\operatorname{am}\left(x,k\right)=\phi,$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E11 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.12		$\operatorname{sn}\left(x,k\right)=\sin\phi=\sin\left(\operatorname{am}\left(x,% k\right)\right),$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\sin\NVar{z}$ : sine function, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E12 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.13		$\operatorname{cn}\left(x,k\right)=\cos\phi=\cos\left(\operatorname{am}\left(x,% k\right)\right).$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cos\NVar{z}$ : cosine function, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E13 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

For $F\left(\phi,k\right)$ see §19.2(ii).

§22.16(ii) Jacobi’s Epsilon Function

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Keywords:: Jacobi’s epsilon function
Notes:: See Lawden (1989, pp. 60–65), Whittaker and Watson (1927, pp. 514–522).
Referenced by:: §22.14(ii), §22.18(i), Erratum (V1.0.1) for Clarifications
Permalink:: http://dlmf.nist.gov/22.16.ii
Clarification (effective with 1.0.1):: The presentation of some of the groups of integral representations in this subsection were rearranged for greater clarity.
See also:: Annotations for §22.16 and Ch.22

Integral Representations

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Keywords:: Jacobian elliptic functions, Jacobi’s epsilon function, definition, integral representations, integrals, of squares
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

For $-K<x<K$ ,

22.16.14

\mathcal{E}\left(x,k\right)=\int_{0}^{\operatorname{sn}\left(x,k\right)}\sqrt{% \frac{1-k^{2}t^{2}}{1-t^{2}}}\,\mathrm{d}t;

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Defines:

\mathcal{E}\left(\NVar{x},\NVar{k}\right)

: Jacobi’s epsilon function

Symbols:

\operatorname{sn}\left(\NVar{z},\NVar{k}\right)

: Jacobian elliptic function,

\,\mathrm{d}\NVar{x}

: differential of

x

\int

: integral,

x

: real and

k

: modulus

Referenced by:

§22.16(ii), Erratum (V1.0.1) for Equation (22.16.14)

Permalink:

http://dlmf.nist.gov/22.16.E14

Encodings:

TeX, pMML, png

Errata (effective with 1.0.1):

Originally appeared with the upper limit as

x

, rather than

\operatorname{sn}\left(x,k\right)

\mathcal{E}\left(x,k\right)=\int_{0}^{x}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}\,% \mathrm{d}t

Reported 2010-07-08 by Charles Karney

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

compare (19.2.5). See Figure 22.16.2.

22.16.15	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-k^{2}\int_{0}^{x}{\operatorname{sn}}^{2}\left(t,k\right)\,% \mathrm{d}t+x,$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus A&S Ref: 16.26.1 Permalink: http://dlmf.nist.gov/22.16.E15 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.16	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=k^{2}\int_{0}^{x}{\operatorname{cn}}^{2}\left(t,k\right)\,% \mathrm{d}t+{k^{\prime}}^{2}x,$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.26.2 Permalink: http://dlmf.nist.gov/22.16.E16 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.17	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=\int_{0}^{x}{\operatorname{dn}}^{2}\left(t,k\right)\,\mathrm{d}t.$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus A&S Ref: 16.26.3 Referenced by: Figure 22.16.2, Figure 22.16.2 Permalink: http://dlmf.nist.gov/22.16.E17 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.18	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-k^{2}\int_{0}^{x}{\operatorname{cd}}^{2}\left(t,k\right)\,% \mathrm{d}t+x+k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k% \right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus A&S Ref: 16.26.4 Permalink: http://dlmf.nist.gov/22.16.E18 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.19	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=k^{2}{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{sd}}^{2}\left(t,% k\right)\,\mathrm{d}t+{k^{\prime}}^{2}x+k^{2}\operatorname{sn}\left(x,k\right)% \operatorname{cd}\left(x,k\right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.26.5 Permalink: http://dlmf.nist.gov/22.16.E19 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.20	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle={k^{\prime}}^{2}\int_{0}^{x}{\operatorname{nd}}^{2}\left(t,k% \right)\,\mathrm{d}t+k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}% \left(x,k\right).$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.26.6 Permalink: http://dlmf.nist.gov/22.16.E20 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

In Equations (22.16.21)–(22.16.23), $-K<x<K.$

22.16.21	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-\int_{0}^{x}{\operatorname{dc}}^{2}\left(t,k\right)\,\mathrm{d}% t+x+\operatorname{sn}\left(x,k\right)\operatorname{dc}\left(x,k\right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus A&S Ref: 16.26.7 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E21 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.22	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{nc}}^{2}\left(t,k% \right)\,\mathrm{d}t+{k^{\prime}}^{2}x+\operatorname{sn}\left(x,k\right)% \operatorname{dc}\left(x,k\right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.26.8 Permalink: http://dlmf.nist.gov/22.16.E22 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.23	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{sc}}^{2}\left(t,k% \right)\,\mathrm{d}t+\operatorname{sn}\left(x,k\right)\operatorname{dc}\left(x% ,k\right).$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.26.9 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E23 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

In Equations (22.16.24)–(22.16.26), $-2K<x<2K$ .

22.16.24	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-\int_{0}^{x}\left({\operatorname{ns}}^{2}\left(t,k\right)-t^{-2% }\right)\,\mathrm{d}t+x^{-1}+x-\operatorname{cn}\left(x,k\right)\operatorname{% ds}\left(x,k\right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus A&S Ref: 16.26.10 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E24 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.25	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-\int_{0}^{x}\left({\operatorname{ds}}^{2}\left(t,k\right)-t^{-2% }\right)\,\mathrm{d}t+x^{-1}+{k^{\prime}}^{2}x-\operatorname{cn}\left(x,k% \right)\operatorname{ds}\left(x,k\right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.26.11 Permalink: http://dlmf.nist.gov/22.16.E25 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.26	$\displaystyle\mathcal{E}\left(x,k\right)$	$\displaystyle=-\int_{0}^{x}\left({\operatorname{cs}}^{2}\left(t,k\right)-t^{-2% }\right)\,\mathrm{d}t+x^{-1}-\operatorname{cn}\left(x,k\right)\operatorname{ds% }\left(x,k\right).$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus A&S Ref: 16.26.12 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E26 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

Quasi-Addition and Quasi-Periodic Formulas

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Keywords:: Jacobi’s epsilon function, Legendre’s elliptic integrals, quasi-addition formula, quasi-periodicity, relations to other functions
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

22.16.27		$\mathcal{E}\left(x_{1}+x_{2},k\right)=\mathcal{E}\left(x_{1},k\right)+\mathcal% {E}\left(x_{2},k\right)-k^{2}\operatorname{sn}\left(x_{1},k\right)% \operatorname{sn}\left(x_{2},k\right)\operatorname{sn}\left(x_{1}+x_{2},k% \right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $x$ : real and $k$ : modulus Referenced by: §22.16(iii) Permalink: http://dlmf.nist.gov/22.16.E27 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.28		$\mathcal{E}\left(x+K,k\right)=\mathcal{E}\left(x,k\right)+E\left(k\right)-k^{2% }\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right),$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E28 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.29		$\mathcal{E}\left(x+2K,k\right)=\mathcal{E}\left(x,k\right)+2E\left(k\right).$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E29 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

For $E\left(k\right)$ see §19.2(ii).

Relation to Theta Functions

ⓘ

Keywords:: Jacobi’s epsilon function, relation to theta functions, relations to other functions, theta functions
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

22.16.30		$\mathcal{E}\left(x,k\right)=\frac{1}{{\theta_{3}}^{2}\left(0,q\right)\theta_{4% }\left(\xi,q\right)}\frac{\mathrm{d}}{\mathrm{d}\xi}\theta_{4}\left(\xi,q% \right)+\frac{E\left(k\right)}{K\left(k\right)}x,$
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $q$ : nome, $x$ : real and $k$ : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E30 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

where $\xi=x/{\theta_{3}}^{2}\left(0,q\right)$ . For $\theta_{j}$ see §20.2(i). For $E\left(k\right)$ see §19.2(ii).

Relation to the Elliptic Integral $E\left(\phi,k\right)$

ⓘ

Keywords:: Jacobi’s epsilon function, relation to Legendre’s elliptic integrals
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

22.16.31		$E\left(\operatorname{am}\left(x,k\right),k\right)=\mathcal{E}\left(x,k\right),$
$-K\leq x\leq K$ .
ⓘ Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E31 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

For $E\left(\phi,k\right)$ see §19.2(ii). See also (22.16.14).

§22.16(iii) Jacobi’s Zeta Function

ⓘ

Keywords:: Jacobi’s zeta function
Notes:: See Lawden (1989, pp. 65–67), Whittaker and Watson (1927, pp. 517–522).
Permalink:: http://dlmf.nist.gov/22.16.iii
See also:: Annotations for §22.16 and Ch.22

Definition

ⓘ

Keywords:: Jacobi’s zeta function, Legendre’s elliptic integrals, definition, relations to other functions
See also:: Annotations for §22.16(iii), §22.16 and Ch.22

With $E\left(k\right)$ and $K\left(k\right)$ as in §19.2(ii) and $x\in\mathbb{R}$ ,

22.16.32		$\mathrm{Z}\left(x\|k\right)=\mathcal{E}\left(x,k\right)-(E\left(k\right)/K\left% (k\right))x.$
ⓘ Defines: $\mathrm{Z}\left(\NVar{x}\|\NVar{k}\right)$ : Jacobi’s zeta function Symbols: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $x$ : real and $k$ : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E32 Encodings: TeX, pMML, png See also: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

See Figure 22.16.3. (Sometimes in the literature $\mathrm{Z}\left(x|k\right)$ is denoted by $\mathrm{Z}(\operatorname{am}\left(x,k\right),k^{2})$ .)

Properties

ⓘ

Keywords:: Jacobi’s zeta function, quasi-addition formula
See also:: Annotations for §22.16(iii), §22.16 and Ch.22

$\mathrm{Z}\left(x|k\right)$ satisfies the same quasi-addition formula as the function $\mathcal{E}\left(x,k\right)$ , given by (22.16.27). Also,

22.16.33		$\mathrm{Z}\left(x+K\|k\right)=\mathrm{Z}\left(x\|k\right)-k^{2}\operatorname{sn}% \left(x,k\right)\operatorname{cd}\left(x,k\right),$
ⓘ Symbols: $\mathrm{Z}\left(\NVar{x}\|\NVar{k}\right)$ : Jacobi’s zeta function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E33 Encodings: TeX, pMML, png See also: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

22.16.34		$\mathrm{Z}\left(x+2K\|k\right)=\mathrm{Z}\left(x\|k\right).$
ⓘ Symbols: $\mathrm{Z}\left(\NVar{x}\|\NVar{k}\right)$ : Jacobi’s zeta function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.16.E34 Encodings: TeX, pMML, png See also: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

§22.16(iv) Graphs

ⓘ

Keywords:: Jacobi’s amplitude function, Jacobi’s epsilon function, Jacobi’s zeta function, graphs
Notes:: Figures 22.16.1–22.16.3 were produced at NIST.
Permalink:: http://dlmf.nist.gov/22.16.iv
See also:: Annotations for §22.16 and Ch.22

See accompanying text — Figure 22.16.1: Jacobi’s amplitude function $\operatorname{am}\left(x,k\right)$ for $0\leq x\leq 10\pi$ and $k=0.4,0.7,0.99,0.999999$ . Values of $k$ greater than 1 are illustrated in Figure 22.19.1. Magnify

22.15 Inverse Functions 22.17 Moduli Outside the Interval [0,1]

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§22.16 Related Functions

Contents

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

Definition

Quasi-Periodicity

Integral Representation

Special Values

Approximation for Small $x$

Approximations for Small $k$ , $k^{\prime}$

Fourier Series

Relation to Elliptic Integrals

§22.16(ii) Jacobi’s Epsilon Function

Integral Representations

Quasi-Addition and Quasi-Periodic Formulas

Relation to Theta Functions

Relation to the Elliptic Integral $E\left(\phi,k\right)$

§22.16(iii) Jacobi’s Zeta Function

Definition

Properties

§22.16(iv) Graphs

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§22.16 Related Functions

Contents

§22.16(i) Jacobi’s Amplitude (am) Function

Definition

Quasi-Periodicity

Integral Representation

Special Values

Approximation for Small x

Approximations for Small k, k′

Fourier Series

Relation to Elliptic Integrals

§22.16(ii) Jacobi’s Epsilon Function

Integral Representations

Quasi-Addition and Quasi-Periodic Formulas

Relation to Theta Functions

Relation to the Elliptic Integral E⁡(ϕ,k)

§22.16(iii) Jacobi’s Zeta Function

Definition

Properties

§22.16(iv) Graphs

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§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

Approximation for Small $x$

Approximations for Small $k$ , $k^{\prime}$

Relation to the Elliptic Integral $E\left(\phi,k\right)$