# §22.16 Related Functions

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

### Definition

 22.16.1 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=\mathop{\mathrm{Arcsin}\/}% \nolimits\!\left(\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\right),$ $x\in\mathbb{R}$, Defines: $\mathop{\mathrm{am}\/}\nolimits\left(\NVar{x},\NVar{k}\right)$: Jacobi’s amplitude function Symbols: $\mathop{\mathrm{sn}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $\in$: element of, $\mathop{\mathrm{Arcsin}\/}\nolimits\NVar{z}$: 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)

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

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

### Integral Representation

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

### Special Values

 22.16.4 $\displaystyle\mathop{\mathrm{am}\/}\nolimits\left(x,0\right)$ $\displaystyle=x,$ Symbols: $\mathop{\mathrm{am}\/}\nolimits\left(\NVar{x},\NVar{k}\right)$: 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) 22.16.5 $\displaystyle\mathop{\mathrm{am}\/}\nolimits\left(x,1\right)$ $\displaystyle=\mathop{\mathrm{gd}\/}\nolimits\!\left(x\right).$

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

### Approximation for Small $x$

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

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

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

### Fourier Series

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

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

### Relation to Elliptic Integrals

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: $\mathop{F\/}\nolimits\!\left(\NVar{\phi},\NVar{k}\right)$: 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)
 22.16.11 $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)=\phi,$ Symbols: $\mathop{\mathrm{am}\/}\nolimits\left(\NVar{x},\NVar{k}\right)$: 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)
 22.16.12 $\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)=\mathop{\sin\/}\nolimits\phi=% \mathop{\sin\/}\nolimits\!\left(\mathop{\mathrm{am}\/}\nolimits\left(x,k\right% )\right),$
 22.16.13 $\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)=\mathop{\cos\/}\nolimits\phi=% \mathop{\cos\/}\nolimits\!\left(\mathop{\mathrm{am}\/}\nolimits\left(x,k\right% )\right).$

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

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

### Integral Representations

For $-K,

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

compare (19.2.5). See Figure 22.16.2.

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

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

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

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

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

 22.16.27 $\mathop{\mathcal{E}\/}\nolimits\!\left(x_{1}+x_{2},k\right)=\mathop{\mathcal{E% }\/}\nolimits\!\left(x_{1},k\right)+\mathop{\mathcal{E}\/}\nolimits\!\left(x_{% 2},k\right)-k^{2}\mathop{\mathrm{sn}\/}\nolimits\left(x_{1},k\right)\mathop{% \mathrm{sn}\/}\nolimits\left(x_{2},k\right)\mathop{\mathrm{sn}\/}\nolimits% \left(x_{1}+x_{2},k\right),$
 22.16.28 $\mathop{\mathcal{E}\/}\nolimits\!\left(x+K,k\right)=\mathop{\mathcal{E}\/}% \nolimits\!\left(x,k\right)+\mathop{E\/}\nolimits\!\left(k\right)-k^{2}\mathop% {\mathrm{sn}\/}\nolimits\left(x,k\right)\mathop{\mathrm{cd}\/}\nolimits\left(x% ,k\right),$
 22.16.29 $\mathop{\mathcal{E}\/}\nolimits\!\left(x+2K,k\right)=\mathop{\mathcal{E}\/}% \nolimits\!\left(x,k\right)+2\!\mathop{E\/}\nolimits\!\left(k\right).$

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

### Relation to Theta Functions

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

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 $\mathop{E\/}\nolimits\!\left(\phi,k\right)$

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

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

## §22.16(iii) Jacobi’s Zeta Function

### Definition

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

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

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

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