§22.16 Related Functions

22.16.1		$$\mathrm{am}(x,k)=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right),$$
$x\in ℝ$,
ⓘ Defines: $\mathrm{am}(x,k)$: Jacobi’s amplitude function Symbols: $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\in$: element of, $\mathrm{Arcsin}z$: general arcsine function, $ℝ$: 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 $-K\le x\le K$ and is defined by continuity elsewhere. See Figure 22.16.1. $\mathrm{am}(x,k)$ is an infinitely differentiable function of $x$.

22.16.2		$$\mathrm{am}(x+2K,k)=\mathrm{am}(x,k)+\pi .$$
ⓘ Symbols: $\mathrm{am}(x,k)$: Jacobi’s amplitude 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, $x$: 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

22.16.3		$$\mathrm{am}(x,k)={\int }_0^x\mathrm{dn}(t,k)dt.$$
ⓘ Symbols: $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: 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 See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.4		$\mathrm{am}(x,0)$	$=x,$
ⓘ Symbols: $\mathrm{am}(x,k)$: 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		$\mathrm{am}(x,1)$	$=\mathrm{gd}\left(x\right).$
ⓘ Symbols: $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: 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 $\mathrm{gd}\left(x\right)$ see §4.23(viii).

22.16.6		$$\mathrm{am}(x,k)=x-k^2\frac{x^3}{3!}+k^2\left(4+k^2\right)\frac{x^5}{5!}+O\left(x^7\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{am}(x,k)$: 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

22.16.7		$$\mathrm{am}(x,k)=x-\frac{1}{4}{k}^2(x-\sin x\cos x)+O\left(k^4\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cos z$: cosine function, $\sin z$: 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		$$\mathrm{am}(x,k)=\mathrm{gd}x-\frac{1}{4}k^{\prime }{}^2(x-\sinh x\cosh x)\mathrm{sech}x+O\left(k^{\prime }{}^4\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: 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

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

22.16.9		$$\mathrm{am}(x,k)=\frac{\pi }{2K}x+2\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\sin \left(2n\zeta \right)}{n(1+q^{2n})}.$$
ⓘ Symbols: $\mathrm{am}(x,k)$: Jacobi’s amplitude 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, $q$: nome, $\sin z$: 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

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

22.16.10		$$x=F(\varphi ,k),$$
ⓘ Symbols: $F(\varphi ,k)$: 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		$$\mathrm{am}(x,k)=\varphi ,$$
ⓘ Symbols: $\mathrm{am}(x,k)$: 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		$$\mathrm{sn}(x,k)=\sin \varphi =\sin \left(\mathrm{am}(x,k)\right),$$
ⓘ Symbols: $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\sin z$: 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		$$\mathrm{cn}(x,k)=\cos \varphi =\cos \left(\mathrm{am}(x,k)\right).$$
ⓘ Symbols: $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: 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 $F(\varphi ,k)$ see §19.2(ii).

For ,

22.16.14		$$ℰ(x,k)={\int }_0^{\mathrm{sn}(x,k)}\sqrt{\frac{1-k^2{t}^2}{1-t^2}}dt;$$
ⓘ Defines: $ℰ(x,k)$: Jacobi’s epsilon function Symbols: $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: 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 $\mathrm{sn}(x,k)$: $$ℰ(x,k)={\int }_0^x\sqrt{\frac{1-k^2{t}^2}{1-t^2}}dt$$ 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		$ℰ(x,k)$	$=-k^2{\displaystyle {\int }_0^x}{\mathrm{sn}}^2(t,k)dt+x,$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{sn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.16		$ℰ(x,k)$	$=k^2{\displaystyle {\int }_0^x}{\mathrm{cn}}^2(t,k)dt+k^{\prime }{}^{2}x,$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.17		$ℰ(x,k)$	$={\displaystyle {\int }_0^x}{\mathrm{dn}}^2(t,k)dt.$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.18		$ℰ(x,k)$	$=-k^2{\displaystyle {\int }_0^x}{\mathrm{cd}}^2(t,k)dt+x+k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.19		$ℰ(x,k)$	$=k^{2}k^{\prime }{}^2{\displaystyle {\int }_0^x}{\mathrm{sd}}^2(t,k)dt+k^{\prime }{}^{2}x+k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.20		$ℰ(x,k)$	$=k^{\prime }{}^2{\displaystyle {\int }_0^x}{\mathrm{nd}}^2(t,k)dt+k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k).$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

In Equations (22.16.21)–(22.16.23),

22.16.21		$ℰ(x,k)$	$=-{\displaystyle {\int }_0^x}{\mathrm{dc}}^2(t,k)dt+x+\mathrm{sn}(x,k)\mathrm{dc}(x,k),$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.22		$ℰ(x,k)$	$=-k^{\prime }{}^2{\displaystyle {\int }_0^x}{\mathrm{nc}}^2(t,k)dt+k^{\prime }{}^{2}x+\mathrm{sn}(x,k)\mathrm{dc}(x,k),$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.23		$ℰ(x,k)$	$=-k^{\prime }{}^2{\displaystyle {\int }_0^x}{\mathrm{sc}}^2(t,k)dt+\mathrm{sn}(x,k)\mathrm{dc}(x,k).$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{sc}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

In Equations (22.16.24)–(22.16.26), .

22.16.24		$ℰ(x,k)$	$=-{\displaystyle {\int }_0^x}\left({\mathrm{ns}}^2(t,k)-t^{-2}\right)dt+x^{-1}+x-\mathrm{cn}(x,k)\mathrm{ds}(x,k),$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.25		$ℰ(x,k)$	$=-{\displaystyle {\int }_0^x}\left({\mathrm{ds}}^2(t,k)-t^{-2}\right)dt+x^{-1}+k^{\prime }{}^{2}x-\mathrm{cn}(x,k)\mathrm{ds}(x,k),$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.26		$ℰ(x,k)$	$=-{\displaystyle {\int }_0^x}\left({\mathrm{cs}}^2(t,k)-t^{-2}\right)dt+x^{-1}-\mathrm{cn}(x,k)\mathrm{ds}(x,k).$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.27		$$ℰ(x_1+x_2,k)=ℰ(x_1,k)+ℰ(x_2,k)-k^2\mathrm{sn}(x_1,k)\mathrm{sn}(x_2,k)\mathrm{sn}(x_1+x_2,k),$$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{sn}(z,k)$: 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		$$ℰ(x+K,k)=ℰ(x,k)+E\left(k\right)-k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $x$: 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		$$ℰ(x+2K,k)=ℰ(x,k)+2E\left(k\right).$$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $x$: 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).

22.16.30		$$ℰ(x,k)=\frac{1}{{\theta }_3^2(0,q){\theta }_4(\xi ,q)}\frac{d}{d\xi }{\theta }_4(\xi ,q)+\frac{E\left(k\right)}{K\left(k\right)}x,$$
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, ${\theta }_j(z,q)$: theta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$: 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 See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

where $\xi =x/{\theta }_3^2(0,q)$. For ${\theta }_j$ see §20.2(i). For $E\left(k\right)$ see §19.2(ii).

22.16.31		$$E(\mathrm{am}(x,k),k)=ℰ(x,k),$$
$-K\le x\le K$.
ⓘ Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E(\varphi ,k)$: 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 $E(\varphi ,k)$ see §19.2(ii). See also (22.16.14).

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

22.16.32		$$\mathrm{Z}\left(x|k\right)=ℰ(x,k)-(E\left(k\right)/K\left(k\right))x.$$
ⓘ Defines: $\mathrm{Z}\left(x|k\right)$: Jacobi’s zeta function Symbols: $ℰ(x,k)$: Jacobi’s epsilon function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $x$: 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 $\mathrm{Z}(\mathrm{am}(x,k),k^2)$.)

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

22.16.33		$$\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k\right)-k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$$
ⓘ Symbols: $\mathrm{Z}\left(x|k\right)$: Jacobi’s zeta function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: 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 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: $\mathrm{Z}\left(x|k\right)$: Jacobi’s zeta function, $K\left(k\right)$: 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 See also: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22