§22.16 Related Functions

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

§22.16(i) Jacobi’s Amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) Function
§22.16(ii) Jacobi’s Epsilon Function
§22.16(iii) Jacobi’s Zeta Function
§22.16(iv) Graphs

§22.16(i) Jacobi’s Amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) Function

Notes:: See Walker (1996, pp. 138–141).
Keywords:: amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function
Referenced by:: §19.25(v), §22.14(i), §29.1, §29.2(ii)
Permalink:: http://dlmf.nist.gov/22.16.i

¶ Definition

Keywords:: amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

22.16.1 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=\mathop{\mathrm{Arcsin}\/}\nolimits\!\left(\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\right),$ $x\in\Real$ ,

Defines:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function
Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\in$ : element of, $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : general arcsine function, $\Real$ : real line (excluding infinity), $x$ : real and $k$ : modulus
Referenced by:: §22.20(vi)
Permalink:: http://dlmf.nist.gov/22.16.E1
Encodings:: TeX, pMML, png

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. $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ is an infinitely differentiable function of $x$ .

¶ Quasi-Periodicity

Keywords:: amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

22.16.2 $\mathop{\mathrm{am}\/}\nolimits\left(x+2K,k\right)=\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)+\pi.$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

¶ Integral Representation

Keywords:: amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

22.16.3 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=\int _{0}^{x}\mathop{\mathrm{dn}\/}\nolimits\left(t,k\right)dt.$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

¶ Special Values

Keywords:: Gudermannian function, amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

22.16.4 $\mathop{\mathrm{am}\/}\nolimits\left(x,0\right)=x,$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function and $x$ : real
Permalink:: http://dlmf.nist.gov/22.16.E4
Encodings:: TeX, pMML, png

22.16.5 $\mathop{\mathrm{am}\/}\nolimits\left(x,1\right)=\mathop{\mathrm{gd}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\mathrm{gd}\/}\nolimits x$ : Gudermannian function, $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function and $x$ : real
Permalink:: http://dlmf.nist.gov/22.16.E5
Encodings:: TeX, pMML, png

For the Gudermannian function $\mathop{\mathrm{gd}\/}\nolimits\!\left(x\right)$ see §4.23(viii).

¶ Approximation for Small $x$

Keywords:: amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

22.16.6 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=x-k^{2}\frac{x^{3}}{3!}+k^{2}\left(4+k^{2}\right)\frac{x^{5}}{5!}+\mathop{O\/}\nolimits\!\left(x^{7}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $!$ : $n!$ : factorial, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E6
Encodings:: TeX, pMML, png

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

Keywords:: amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

22.16.7 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=x-\tfrac{1}{4}k^{2}(x-\mathop{\sin\/}\nolimits x\mathop{\cos\/}\nolimits x)+\mathop{O\/}\nolimits\!\left(k^{4}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

22.16.8 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=\mathop{\mathrm{gd}\/}\nolimits x-\tfrac{1}{4}{k^{{\prime}}}^{2}(x-\mathop{\sinh\/}\nolimits x\mathop{\cosh\/}\nolimits x)\mathop{\mathrm{sech}\/}\nolimits x+\mathop{O\/}\nolimits\left({k^{{\prime}}}^{4}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{gd}\/}\nolimits x$ : Gudermannian function, $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : 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

¶ Fourier Series

Keywords:: Fourier series, amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

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

22.16.9 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=\frac{\pi}{2K}x+2\sum _{{n=1}}^{{\infty}}\frac{q^{n}\mathop{\sin\/}\nolimits\!\left(2n\zeta\right)}{n(1+q^{{2n}})}.$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $q$ : nome, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real and $k$ : modulus
Referenced by:: §22.20(vi)
Permalink:: http://dlmf.nist.gov/22.16.E9
Encodings:: TeX, pMML, png

¶ Relation to Elliptic Integrals

Keywords:: Legendre’s elliptic integrals, amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function

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

22.16.10 $x=\mathop{F\/}\nolimits\!\left(\phi,k\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : 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

22.16.11 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=\phi,$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E11
Encodings:: TeX, pMML, png

22.16.12 $\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)=\mathop{\sin\/}\nolimits\phi=\mathop{\sin\/}\nolimits\!\left(\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E12
Encodings:: TeX, pMML, png

22.16.13 $\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)=\mathop{\cos\/}\nolimits\phi=\mathop{\cos\/}\nolimits\!\left(\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)\right).$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\cos\/}\nolimits z$ : cosine function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E13
Encodings:: TeX, pMML, png

For $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ see §19.2(ii).

§22.16(ii) Jacobi’s Epsilon Function

Notes:: See Lawden (1989, pp. 60–65), Whittaker and Watson (1927, pp. 514–522).
Keywords:: Jacobi’s epsilon function
Referenced by:: §22.14(ii), ¶ ‣ §22.18(i), Version 1.0.1 (June 27, 2011)
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.
Reported 2011-06-21

¶ Integral Representations

Keywords:: Jacobian elliptic functions, Jacobi’s epsilon function, integrals

For $-K<x<K$ ,

22.16.14 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=\int _{0}^{{\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)}}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt;$

Defines:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function
Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus
Referenced by:: ¶ ‣ §22.16(ii), Version 1.0.1 (June 27, 2011)
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 $\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)$ :
$\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=\int _{0}^{x}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt$

Reported 2010-07-08 by Charles Karney

compare (19.2.5). See Figure 22.16.2.

22.16.15 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-k^{2}\int _{0}^{x}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(t,k\right)dt+x,$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.16 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=k^{2}\int _{0}^{x}{\mathop{\mathrm{cn}\/}\nolimits^{{2}}}\left(t,k\right)dt+{k^{{\prime}}}^{2}x,$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.17 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=\int _{0}^{x}{\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(t,k\right)dt.$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.18 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-k^{2}\int _{0}^{x}{\mathop{\mathrm{cd}\/}\nolimits^{{2}}}\left(t,k\right)dt+x+k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.19 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=k^{2}{k^{{\prime}}}^{2}\int _{0}^{x}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}\left(t,k\right)dt+{k^{{\prime}}}^{2}x+k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.20 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)={k^{{\prime}}}^{2}\int _{0}^{x}{\mathop{\mathrm{nd}\/}\nolimits^{{2}}}\left(t,k\right)dt+k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right).$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

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

22.16.21 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-\int _{0}^{x}{\mathop{\mathrm{dc}\/}\nolimits^{{2}}}\left(t,k\right)dt+x+\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{dc}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.22 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-{k^{{\prime}}}^{2}\int _{0}^{x}{\mathop{\mathrm{nc}\/}\nolimits^{{2}}}\left(t,k\right)dt+{k^{{\prime}}}^{2}x+\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{dc}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.23 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-{k^{{\prime}}}^{2}\int _{0}^{x}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(t,k\right)dt+\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{dc}\/}\nolimits\left(x,k\right).$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

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

22.16.24 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-\int _{0}^{x}\left({\mathop{\mathrm{ns}\/}\nolimits^{{2}}}\left(t,k\right)-t^{{-2}}\right)dt+x^{{-1}}+x-\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{ds}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.25 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-\int _{0}^{x}\left({\mathop{\mathrm{ds}\/}\nolimits^{{2}}}\left(t,k\right)-t^{{-2}}\right)dt+x^{{-1}}+{k^{{\prime}}}^{2}x-\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{ds}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

22.16.26 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=-\int _{0}^{x}\left({\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(t,k\right)-t^{{-2}}\right)dt+x^{{-1}}-\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{ds}\/}\nolimits\left(x,k\right).$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

¶ Quasi-Addition and Quasi-Periodic Formulas

Keywords:: Jacobi’s epsilon function, Legendre’s elliptic integrals

22.16.27 $\mathop{\mathcal{E}\/}\nolimits\!\left(x_{1}+x_{2},k\right)=\mathop{\mathcal{E}\/}\nolimits\!\left(x_{1},k\right)+\mathop{\mathcal{E}\/}\nolimits\!\left(x_{2},k\right)-k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(x_{1},k\right)\mathop{\mathrm{sn}\/}\nolimits\left(x_{2},k\right)\mathop{\mathrm{sn}\/}\nolimits\left(x_{1}+x_{2},k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : 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

22.16.28 $\mathop{\mathcal{E}\/}\nolimits\!\left(x+K,k\right)=\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)+\mathop{E\/}\nolimits\!\left(k\right)-k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : 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

22.16.29 $\mathop{\mathcal{E}\/}\nolimits\!\left(x+2K,k\right)=\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)+2\!\mathop{E\/}\nolimits\!\left(k\right).$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : 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

For $\mathop{E\/}\nolimits\!\left(k\right)$ see §19.2(ii).

¶ Relation to Theta Functions

Keywords:: Jacobi’s epsilon function, theta functions

22.16.30 $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=\frac{1}{{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(\xi,q\right)}\frac{d}{d\xi}\mathop{\theta _{{4}}\/}\nolimits\!\left(\xi,q\right)+\frac{\mathop{E\/}\nolimits\!\left(k\right)}{\mathop{K\/}\nolimits\!\left(k\right)}x,$

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$ : 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

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

¶ Relation to the Elliptic Integral $\mathop{E\/}\nolimits\!\left(\phi,k\right)$

Keywords:: Jacobi’s epsilon function

22.16.31 $\mathop{E\/}\nolimits\!\left(\mathop{\mathrm{am}\/}\nolimits\left(x,k\right),k\right)=\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right),$ $-K\leq x\leq K$ .

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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

For $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ see §19.2(ii). See also (22.16.14).

§22.16(iii) Jacobi’s Zeta Function

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

¶ Definition

Keywords:: Jacobi’s zeta function, Legendre’s elliptic integrals

With $\mathop{E\/}\nolimits\!\left(k\right)$ and $\mathop{K\/}\nolimits\!\left(k\right)$ as in §19.2(ii) and $x\in\Real$ ,

22.16.32 $\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)=\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)-(\mathop{E\/}\nolimits\!\left(k\right)/\mathop{K\/}\nolimits\!\left(k\right))x.$

Defines:: $\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)$ : Jacobi’s zeta function
Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : 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 Figure 22.16.3. (Sometimes in the literature $\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)$ is denoted by $\mathop{\mathrm{Z}\/}\nolimits(\mathop{\mathrm{am}\/}\nolimits\left(x,k\right),k^{2})$ .)

¶ Properties

Keywords:: Jacobi’s zeta function

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

22.16.33 $\mathop{\mathrm{Z}\/}\nolimits\!\left(x+K|k\right)=\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)-k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right),$

Symbols:: $\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)$ : Jacobi’s zeta function, $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

22.16.34 $\mathop{\mathrm{Z}\/}\nolimits\!\left(x+2K|k\right)=\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right).$

Symbols:: $\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)$ : Jacobi’s zeta function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

§22.16(iv) Graphs

Notes:: Figures 22.16.1–22.16.3 were produced at NIST.
Keywords:: Jacobi’s amplitude function, Jacobi’s epsilon function, Jacobi’s zeta function
Permalink:: http://dlmf.nist.gov/22.16.iv

Figure 22.16.1: Jacobi’s amplitude function $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ 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.

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $x$ : real and $k$ : modulus
Referenced by:: Figure 22.16.2, Figure 22.16.2, ¶ ‣ §22.16(i), §22.16(iv), Figure 22.19.1, Figure 22.19.1, §22.19(i)
Permalink:: http://dlmf.nist.gov/22.16.F1
Encodings:: pdf, png

Figure 22.16.2: Jacobi’s epsilon function $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ for $0\leq x\leq 10\pi$ and $k=0.4,0.7,0.99,0.999999$ . (These graphs are similar to those in Figure 22.16.1; compare (22.16.3), (22.16.17), and the graphs of $\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)$ in §22.3(i).)

Symbols:: $\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$ : Jacobi’s epsilon function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $x$ : real and $k$ : modulus
Referenced by:: ¶ ‣ §22.16(ii)
Permalink:: http://dlmf.nist.gov/22.16.F2
Encodings:: pdf, png

Figure 22.16.3: Jacobi’s zeta function $\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)$ for $0\leq x\leq 10\pi$ and $k=0.4,0.7,0.99,0.999999$ .

Symbols:: $\mathop{\mathrm{Z}\/}\nolimits\!\left(x|k\right)$ : Jacobi’s zeta function, $x$ : real and $k$ : modulus
Referenced by:: ¶ ‣ §22.16(iii), §22.16(iv)
Permalink:: http://dlmf.nist.gov/22.16.F3
Encodings:: pdf, png

§22.16 Related Functions

Contents

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

¶ Definition

¶ Quasi-Periodicity

¶ Integral Representation

¶ Special Values

¶ Approximation for Small

¶ Approximations for Small ,

¶ 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

§22.16(iii) Jacobi’s Zeta Function

¶ Definition

¶ Properties

§22.16(iv) Graphs

§22.16(i) Jacobi’s Amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) Function

¶ Approximation for Small $x$

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

¶ Relation to the Elliptic Integral $\mathop{E\/}\nolimits\!\left(\phi,k\right)$