# §22.15 Inverse Functions

## §22.15(i) Definitions

The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). With real variables, the solutions of the equations

 22.15.1 $\operatorname{sn}\left(\xi,k\right)=x,$ $-1\leq x\leq 1$, ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $x$: real, $k$: modulus and $\xi$: solution Referenced by: §22.15(i), §22.15(i) Permalink: http://dlmf.nist.gov/22.15.E1 Encodings: TeX, pMML, png See also: Annotations for 22.15(i), 22.15 and 22
 22.15.2 $\operatorname{cn}\left(\eta,k\right)=x,$ $-1\leq x\leq 1$, ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $x$: real, $k$: modulus and $\eta$: solution Referenced by: §22.15(i) Permalink: http://dlmf.nist.gov/22.15.E2 Encodings: TeX, pMML, png See also: Annotations for 22.15(i), 22.15 and 22
 22.15.3 $\operatorname{dn}\left(\zeta,k\right)=x,$ $k^{\prime}\leq x\leq 1$, ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $x$: real, $k$: modulus, $k^{\prime}$: complementary modulus and $\zeta$: solution Referenced by: §22.15(i) Permalink: http://dlmf.nist.gov/22.15.E3 Encodings: TeX, pMML, png See also: Annotations for 22.15(i), 22.15 and 22

are denoted respectively by

 22.15.4 $\displaystyle\xi$ $\displaystyle=\operatorname{arcsn}\left(x,k\right),$ $\displaystyle\eta$ $\displaystyle=\operatorname{arccn}\left(x,k\right),$ $\displaystyle\zeta$ $\displaystyle=\operatorname{arcdn}\left(x,k\right).$ ⓘ Defines: $\xi$: solution (locally), $\eta$: solution (locally) and $\zeta$: solution (locally) Symbols: $\operatorname{arccn}\left(\NVar{x},\NVar{k}\right)$: inverse Jacobian elliptic function, $\operatorname{arcdn}\left(\NVar{x},\NVar{k}\right)$: inverse Jacobian elliptic function, $\operatorname{arcsn}\left(\NVar{x},\NVar{k}\right)$: inverse Jacobian elliptic function, $x$: real and $k$: modulus Referenced by: §22.15(i) Permalink: http://dlmf.nist.gov/22.15.E4 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 22.15(i), 22.15 and 22

Each of these inverse functions is multivalued. The principal values satisfy

 22.15.5 $\displaystyle-K$ $\displaystyle\leq\operatorname{arcsn}\left(x,k\right)\leq K,$ 22.15.6 $\displaystyle 0$ $\displaystyle\leq\operatorname{arccn}\left(x,k\right)\leq 2K,$ 22.15.7 $\displaystyle 0$ $\displaystyle\leq\operatorname{arcdn}\left(x,k\right)\leq K,$

and unless stated otherwise it is assumed that the inverse functions assume their principal values. The general solutions of (22.15.1), (22.15.2), (22.15.3) are, respectively,

 22.15.8 $\displaystyle\xi$ $\displaystyle=(-1)^{m}\operatorname{arcsn}\left(x,k\right)+2mK,$ 22.15.9 $\displaystyle\eta$ $\displaystyle=\pm\operatorname{arccn}\left(x,k\right)+4mK,$ 22.15.10 $\displaystyle\zeta$ $\displaystyle=\pm\operatorname{arcdn}\left(x,k\right)+2mK,$

where $m\in\mathbb{Z}$.

Equations (22.15.1) and (22.15.4), for $\operatorname{arcsn}\left(x,k\right)$, are equivalent to (22.15.12) and also to

 22.15.11 $x=\int_{0}^{\operatorname{sn}\left(x,k\right)}\frac{\mathrm{d}t}{\sqrt{(1-t^{2% })(1-k^{2}t^{2})}},$ $-1\leq x\leq 1$, $0\leq k\leq 1$.

Similarly with (22.15.13)–(22.15.23) and the other eleven Jacobian elliptic functions.

## §22.15(ii) Representations as Elliptic Integrals

 22.15.12 $\operatorname{arcsn}\left(x,k\right)=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t% ^{2})(1-k^{2}t^{2})}},$ $-1\leq x\leq 1$, ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\operatorname{arcsn}\left(\NVar{x},\NVar{k}\right)$: inverse Jacobian elliptic function, $x$: real and $k$: modulus A&S Ref: 17.4.45 Referenced by: §22.15(i), §22.15(ii) Permalink: http://dlmf.nist.gov/22.15.E12 Encodings: TeX, pMML, png See also: Annotations for 22.15(ii), 22.15 and 22
 22.15.13 $\operatorname{arccn}\left(x,k\right)=\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t% ^{2})({k^{\prime}}^{2}+k^{2}t^{2})}},$ $-1\leq x\leq 1$,
 22.15.14 $\operatorname{arcdn}\left(x,k\right)=\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t% ^{2})(t^{2}-{k^{\prime}}^{2})}},$ $k^{\prime}\leq x\leq 1$ .
 22.15.15 $\operatorname{arccd}\left(x,k\right)=\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t% ^{2})(1-k^{2}t^{2})}},$ $-1\leq x\leq 1$,
 22.15.16 $\operatorname{arcsd}\left(x,k\right)=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-{% k^{\prime}}^{2}t^{2})(1+k^{2}t^{2})}},$ $-1/k^{\prime}\leq x\leq 1/k^{\prime}$,
 22.15.17 $\operatorname{arcnd}\left(x,k\right)=\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{(t^{% 2}-1)(1-{k^{\prime}}^{2}t^{2})}},$ $1\leq x\leq 1/k^{\prime}$,
 22.15.18 $\operatorname{arcdc}\left(x,k\right)=\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{(t^{% 2}-1)(t^{2}-k^{2})}},$ $1\leq x<\infty$,
 22.15.19 $\operatorname{arcnc}\left(x,k\right)=\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{(t^{% 2}-1)(k^{2}+{k^{\prime}}^{2}t^{2})}},$ $1\leq x<\infty$,
 22.15.20 $\operatorname{arcsc}\left(x,k\right)=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1+t% ^{2})(1+{k^{\prime}}^{2}t^{2})}},$ $-\infty,
 22.15.21 $\operatorname{arcns}\left(x,k\right)=\int_{x}^{\infty}\frac{\mathrm{d}t}{\sqrt% {(t^{2}-1)(t^{2}-k^{2})}},$ $1\leq x<\infty$,
 22.15.22 $\operatorname{arcds}\left(x,k\right)=\int_{x}^{\infty}\frac{\mathrm{d}t}{\sqrt% {(t^{2}+k^{2})(t^{2}-{k^{\prime}}^{2})}},$ $k^{\prime}\leq x<\infty$,
 22.15.23 $\operatorname{arccs}\left(x,k\right)=\int_{x}^{\infty}\frac{\mathrm{d}t}{\sqrt% {(1+t^{2})(t^{2}+{k^{\prime}}^{2})}},$ $-\infty.

The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. Other integrals, for example,

 $\int_{x}^{b}\frac{\mathrm{d}t}{\sqrt{(a^{2}+t^{2})(b^{2}-t^{2})}}$

can be transformed into normal form by elementary change of variables. Comprehensive treatments are given by Carlson (2005), Lawden (1989, pp. 52–55), Bowman (1953, Chapter IX), and Erdélyi et al. (1953b, pp. 296–301). See also Abramowitz and Stegun (1964, p. 596).

For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). For power-series expansions see Carlson (2008).