22 Jacobian Elliptic Functions Properties 22.14 Integrals 22.16 Related Functions

§22.15 Inverse Functions

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Keywords:: Jacobian elliptic functions, inverse, inverse Jacobian elliptic functions
Referenced by:: §19.25(v)
Permalink:: http://dlmf.nist.gov/22.15
See also:: Annotations for Ch.22

§22.15(i) Definitions
§22.15(ii) Representations as Elliptic Integrals

§22.15(i) Definitions

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Defines:: $\operatorname{arccd}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arccn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arccs}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcdc}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcdn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcds}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcnc}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcnd}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcns}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcsc}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $\operatorname{arcsd}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function and $\operatorname{arcsn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function
Keywords:: definitions, inverse Jacobian elliptic functions, notation, principal values
Notes:: See Lawden (1989, pp. 50–55), Bowman (1953, Chapter 1), Whittaker and Watson (1927, pp. 514–517).
Permalink:: http://dlmf.nist.gov/22.15.i
See also:: Annotations for §22.15 and Ch.22

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 Ch.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 Ch.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 Ch.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 Ch.22

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

22.15.5	$\displaystyle-K$	$\displaystyle\leq\operatorname{arcsn}\left(x,k\right)\leq K,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arcsn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.15.E5 Encodings: TeX, pMML, png See also: Annotations for §22.15(i), §22.15 and Ch.22
22.15.6	$\displaystyle 0$	$\displaystyle\leq\operatorname{arccn}\left(x,k\right)\leq 2K,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arccn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.15.E6 Encodings: TeX, pMML, png See also: Annotations for §22.15(i), §22.15 and Ch.22
22.15.7	$\displaystyle 0$	$\displaystyle\leq\operatorname{arcdn}\left(x,k\right)\leq K,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arcdn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.15.E7 Encodings: TeX, pMML, png See also: Annotations for §22.15(i), §22.15 and Ch.22

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	$\displaystyle\xi$	$\displaystyle=(-1)^{m}\operatorname{arcsn}\left(x,k\right)+2mK,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arcsn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus, $\xi$ : solution and $m$ : integer Permalink: http://dlmf.nist.gov/22.15.E8 Encodings: TeX, pMML, png See also: Annotations for §22.15(i), §22.15 and Ch.22
22.15.9	$\displaystyle\eta$	$\displaystyle=\pm\operatorname{arccn}\left(x,k\right)+4mK,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arccn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus, $\eta$ : solution and $m$ : integer Permalink: http://dlmf.nist.gov/22.15.E9 Encodings: TeX, pMML, png See also: Annotations for §22.15(i), §22.15 and Ch.22
22.15.10	$\displaystyle\zeta$	$\displaystyle=\pm\operatorname{arcdn}\left(x,k\right)+2mK,$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arcdn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus, $\zeta$ : solution and $m$ : integer Permalink: http://dlmf.nist.gov/22.15.E10 Encodings: TeX, pMML, png See also: Annotations for §22.15(i), §22.15 and Ch.22

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		$x=\int_{0}^{\operatorname{sn}\left(x,k\right)}\frac{\,\mathrm{d}t}{\sqrt{(1-t^% {2})(1-k^{2}t^{2})}},$
$-1\leq x\leq 1$ , $0\leq k\leq 1$ .
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus Referenced by: §19.25(v) Permalink: http://dlmf.nist.gov/22.15.E11 Encodings: TeX, pMML, png See also: Annotations for §22.15(i), §22.15 and Ch.22

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

§22.15(ii) Representations as Elliptic Integrals

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Keywords:: Legendre’s elliptic integrals, as Legendre’s elliptic integrals, as symmetric elliptic integrals, equivalent forms, inverse Jacobian elliptic functions, normal forms, power-series expansions, relations to other functions
Notes:: See Lawden (1989, pp. 50–55) and Bowman (1953, Chapter 1). (The conditions in the former are unnecessarily restrictive.)
Referenced by:: §19.10(i), §22.10(ii), §32.10(vi)
Permalink:: http://dlmf.nist.gov/22.15.ii
See also:: Annotations for §22.15 and Ch.22

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 Ch.22

22.15.13		$\operatorname{arccn}\left(x,k\right)=\int_{x}^{1}\frac{\,\mathrm{d}t}{\sqrt{(1% -t^{2})({k^{\prime}}^{2}+k^{2}t^{2})}},$
$-1\leq x\leq 1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.52 Referenced by: §22.15(i) Permalink: http://dlmf.nist.gov/22.15.E13 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.14		$\operatorname{arcdn}\left(x,k\right)=\int_{x}^{1}\frac{\,\mathrm{d}t}{\sqrt{(1% -t^{2})(t^{2}-{k^{\prime}}^{2})}},$
$k^{\prime}\leq x\leq 1$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcdn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.44 Referenced by: §22.15(ii) Permalink: http://dlmf.nist.gov/22.15.E14 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.15		$\operatorname{arccd}\left(x,k\right)=\int_{x}^{1}\frac{\,\mathrm{d}t}{\sqrt{(1% -t^{2})(1-k^{2}t^{2})}},$
$-1\leq x\leq 1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccd}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus A&S Ref: 17.4.46 Permalink: http://dlmf.nist.gov/22.15.E15 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.16		$\operatorname{arcsd}\left(x,k\right)=\int_{0}^{x}\frac{\,\mathrm{d}t}{\sqrt{(1% -{k^{\prime}}^{2}t^{2})(1+k^{2}t^{2})}},$
$-1/k^{\prime}\leq x\leq 1/k^{\prime}$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcsd}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.51 Permalink: http://dlmf.nist.gov/22.15.E16 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.17		$\operatorname{arcnd}\left(x,k\right)=\int_{1}^{x}\frac{\,\mathrm{d}t}{\sqrt{(t% ^{2}-1)(1-{k^{\prime}}^{2}t^{2})}},$
$1\leq x\leq 1/k^{\prime}$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcnd}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.43 Permalink: http://dlmf.nist.gov/22.15.E17 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.18		$\operatorname{arcdc}\left(x,k\right)=\int_{1}^{x}\frac{\,\mathrm{d}t}{\sqrt{(t% ^{2}-1)(t^{2}-k^{2})}},$
$1\leq x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcdc}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus A&S Ref: 17.4.47 Permalink: http://dlmf.nist.gov/22.15.E18 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.19		$\operatorname{arcnc}\left(x,k\right)=\int_{1}^{x}\frac{\,\mathrm{d}t}{\sqrt{(t% ^{2}-1)(k^{2}+{k^{\prime}}^{2}t^{2})}},$
$1\leq x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcnc}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.49 Permalink: http://dlmf.nist.gov/22.15.E19 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.20		$\operatorname{arcsc}\left(x,k\right)=\int_{0}^{x}\frac{\,\mathrm{d}t}{\sqrt{(1% +t^{2})(1+{k^{\prime}}^{2}t^{2})}},$
$-\infty<x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcsc}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.41 Permalink: http://dlmf.nist.gov/22.15.E20 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.21		$\operatorname{arcns}\left(x,k\right)=\int_{x}^{\infty}\frac{\,\mathrm{d}t}{% \sqrt{(t^{2}-1)(t^{2}-k^{2})}},$
$1\leq x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcns}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus A&S Ref: 17.4.48 Permalink: http://dlmf.nist.gov/22.15.E21 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.22		$\operatorname{arcds}\left(x,k\right)=\int_{x}^{\infty}\frac{\,\mathrm{d}t}{% \sqrt{(t^{2}+k^{2})(t^{2}-{k^{\prime}}^{2})}},$
$k^{\prime}\leq x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcds}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.50 Permalink: http://dlmf.nist.gov/22.15.E22 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

22.15.23		$\operatorname{arccs}\left(x,k\right)=\int_{x}^{\infty}\frac{\,\mathrm{d}t}{% \sqrt{(1+t^{2})(t^{2}+{k^{\prime}}^{2})}},$
$-\infty<x<\infty$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccs}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 17.4.42 Referenced by: §22.15(i) Permalink: http://dlmf.nist.gov/22.15.E23 Encodings: TeX, pMML, png See also: Annotations for §22.15(ii), §22.15 and Ch.22

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).

22.14 Integrals 22.16 Related Functions

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§22.15 Inverse Functions

Contents

§22.15(i) Definitions

§22.15(ii) Representations as Elliptic Integrals