# §22.16 Related Functions

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

### Definition

 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

 22.16.2 $\operatorname{am}\left(x+2K,k\right)=\operatorname{am}\left(x,k\right)+\pi.$

### Integral Representation

 22.16.3 $\operatorname{am}\left(x,k\right)=\int_{0}^{x}\operatorname{dn}\left(t,k\right% )\,\mathrm{d}t.$

### Special Values

 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 $x$

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

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

 22.16.7 $\operatorname{am}\left(x,k\right)=x-\tfrac{1}{4}k^{2}(x-\sin x\cos x)+O\left(k% ^{4}\right),$
 22.16.8 $\operatorname{am}\left(x,k\right)=\operatorname{gd}x-\tfrac{1}{4}{k^{\prime}}^% {2}(x-\sinh x\cosh x)\operatorname{sech}x+O\left({k^{\prime}}^{4}\right).$

### Fourier Series

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

 22.16.9 $\operatorname{am}\left(x,k\right)=\frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^% {n}\sin\left(2n\zeta\right)}{n(1+q^{2n})}.$

### Relation to Elliptic Integrals

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 $\operatorname{sn}\left(x,k\right)=\sin\phi=\sin\left(\operatorname{am}\left(x,% k\right)\right),$
 22.16.13 $\operatorname{cn}\left(x,k\right)=\cos\phi=\cos\left(\operatorname{am}\left(x,% k\right)\right).$

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

## §22.16(ii) Jacobi’s Epsilon Function

### Integral Representations

For $-K,

22.16.14 $\mathcal{E}\left(x,k\right)=\int_{0}^{\operatorname{sn}\left(x,k\right)}\sqrt{% \frac{1-k^{2}t^{2}}{1-t^{2}}}\,\mathrm{d}t;$

compare (19.2.5). See Figure 22.16.2.

 22.16.15 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-k^{2}\int_{0}^{x}{\operatorname{sn}}^{2}\left(t,k\right)\,% \mathrm{d}t+x,$ 22.16.16 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=k^{2}\int_{0}^{x}{\operatorname{cn}}^{2}\left(t,k\right)\,% \mathrm{d}t+{k^{\prime}}^{2}x,$ 22.16.17 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=\int_{0}^{x}{\operatorname{dn}}^{2}\left(t,k\right)\,\mathrm{d}t.$
 22.16.18 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-k^{2}\int_{0}^{x}{\operatorname{cd}}^{2}\left(t,k\right)\,% \mathrm{d}t+x+k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k% \right),$ 22.16.19 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=k^{2}{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{sd}}^{2}\left(t,% k\right)\,\mathrm{d}t+{k^{\prime}}^{2}x+k^{2}\operatorname{sn}\left(x,k\right)% \operatorname{cd}\left(x,k\right),$ 22.16.20 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle={k^{\prime}}^{2}\int_{0}^{x}{\operatorname{nd}}^{2}\left(t,k% \right)\,\mathrm{d}t+k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}% \left(x,k\right).$

In Equations (22.16.21)–(22.16.23), $-K

 22.16.21 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-\int_{0}^{x}{\operatorname{dc}}^{2}\left(t,k\right)\,\mathrm{d}% t+x+\operatorname{sn}\left(x,k\right)\operatorname{dc}\left(x,k\right),$ 22.16.22 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{nc}}^{2}\left(t,k% \right)\,\mathrm{d}t+{k^{\prime}}^{2}x+\operatorname{sn}\left(x,k\right)% \operatorname{dc}\left(x,k\right),$ 22.16.23 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{sc}}^{2}\left(t,k% \right)\,\mathrm{d}t+\operatorname{sn}\left(x,k\right)\operatorname{dc}\left(x% ,k\right).$

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

 22.16.24 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-\int_{0}^{x}\left({\operatorname{ns}}^{2}\left(t,k\right)-t^{-2% }\right)\,\mathrm{d}t+x^{-1}+x-\operatorname{cn}\left(x,k\right)\operatorname{% ds}\left(x,k\right),$ 22.16.25 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-\int_{0}^{x}\left({\operatorname{ds}}^{2}\left(t,k\right)-t^{-2% }\right)\,\mathrm{d}t+x^{-1}+{k^{\prime}}^{2}x-\operatorname{cn}\left(x,k% \right)\operatorname{ds}\left(x,k\right),$ 22.16.26 $\displaystyle\mathcal{E}\left(x,k\right)$ $\displaystyle=-\int_{0}^{x}\left({\operatorname{cs}}^{2}\left(t,k\right)-t^{-2% }\right)\,\mathrm{d}t+x^{-1}-\operatorname{cn}\left(x,k\right)\operatorname{ds% }\left(x,k\right).$

 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 $\mathcal{E}\left(x+K,k\right)=\mathcal{E}\left(x,k\right)+E\left(k\right)-k^{2% }\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right),$
 22.16.29 $\mathcal{E}\left(x+2K,k\right)=\mathcal{E}\left(x,k\right)+2E\left(k\right).$

For $E\left(k\right)$ see §19.2(ii).

### Relation to Theta Functions

 22.16.30 $\mathcal{E}\left(x,k\right)=\frac{1}{{\theta_{3}}^{2}\left(0,q\right)\theta_{4% }\left(\xi,q\right)}\frac{\mathrm{d}}{\mathrm{d}\xi}\theta_{4}\left(\xi,q% \right)+\frac{E\left(k\right)}{K\left(k\right)}x,$

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 $E\left(\phi,k\right)$

 22.16.31 $E\left(\operatorname{am}\left(x,k\right),k\right)=\mathcal{E}\left(x,k\right),$ $-K\leq x\leq K$.

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

## §22.16(iii) Jacobi’s Zeta Function

### Definition

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

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

 22.16.33 $\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k\right)-k^{2}\operatorname{sn}% \left(x,k\right)\operatorname{cd}\left(x,k\right),$
 22.16.34 $\mathrm{Z}\left(x+2K|k\right)=\mathrm{Z}\left(x|k\right).$