22 Jacobian Elliptic Functions Properties 22.13 Derivatives and Differential Equations 22.15 Inverse Functions

§22.14 Integrals

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Permalink:: http://dlmf.nist.gov/22.14
See also:: Annotations for Ch.22

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions
§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions
§22.14(iii) Other Indefinite Integrals
§22.14(iv) Definite Integrals

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

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Keywords:: Jacobian elliptic functions, indefinite, integrals
Notes:: See Lawden (1989, pp. 40–41), Whittaker and Watson (1927, pp. 514–517).
Permalink:: http://dlmf.nist.gov/22.14.i
See also:: Annotations for §22.14 and Ch.22

With $x\in\mathbb{R}$ ,

22.14.1	$\displaystyle\int\operatorname{sn}\left(x,k\right)\,\mathrm{d}x$	$\displaystyle=k^{-1}\ln\left(\operatorname{dn}\left(x,k\right)-k\operatorname{% cn}\left(x,k\right)\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus A&S Ref: 16.24.1 Permalink: http://dlmf.nist.gov/22.14.E1 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22
22.14.2	$\displaystyle\int\operatorname{cn}\left(x,k\right)\,\mathrm{d}x$	$\displaystyle=k^{-1}\operatorname{Arccos}\left(\operatorname{dn}\left(x,k% \right)\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arccos}\NVar{z}$ : general arccosine function, $x$ : real and $k$ : modulus A&S Ref: 16.24.2 Permalink: http://dlmf.nist.gov/22.14.E2 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22
22.14.3	$\displaystyle\int\operatorname{dn}\left(x,k\right)\,\mathrm{d}x$	$\displaystyle=\operatorname{Arcsin}\left(\operatorname{sn}\left(x,k\right)% \right)=\operatorname{am}\left(x,k\right).$
ⓘ Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $x$ : real and $k$ : modulus A&S Ref: 16.24.3 Permalink: http://dlmf.nist.gov/22.14.E3 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

The branches of the inverse trigonometric functions are chosen so that they are continuous. See §22.16(i) for $\operatorname{am}\left(z,k\right)$ .

Secondly,

22.14.4		$\int\operatorname{cd}\left(x,k\right)\,\mathrm{d}x=k^{-1}\ln\left(% \operatorname{nd}\left(x,k\right)+k\operatorname{sd}\left(x,k\right)\right),$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus A&S Ref: 16.24.4 Permalink: http://dlmf.nist.gov/22.14.E4 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

22.14.5		$\int\operatorname{sd}\left(x,k\right)\,\mathrm{d}x=(kk^{\prime})^{-1}% \operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right),$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.24.5 Permalink: http://dlmf.nist.gov/22.14.E5 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

22.14.6		$\int\operatorname{nd}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}% \operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)\right).$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arccos}\NVar{z}$ : general arccosine function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.24.6 Permalink: http://dlmf.nist.gov/22.14.E6 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

Again, the branches of the inverse trigonometric functions must be continuous.

Thirdly, with $-K<x<K$ ,

22.14.7		$\int\operatorname{dc}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{nc}% \left(x,k\right)+\operatorname{sc}\left(x,k\right)\right),$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus A&S Ref: 16.24.7 Permalink: http://dlmf.nist.gov/22.14.E7 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

22.14.8		$\int\operatorname{nc}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}\ln\left(% \operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{sc}\left(x,k\right)% \right),$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.24.8 Permalink: http://dlmf.nist.gov/22.14.E8 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

22.14.9		$\int\operatorname{sc}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}\ln\left(% \operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{nc}\left(x,k\right)% \right).$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.24.9 Permalink: http://dlmf.nist.gov/22.14.E9 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

Lastly, with $0<x<2K$ ,

22.14.10		$\int\operatorname{ns}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ds}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus A&S Ref: 16.24.10 Permalink: http://dlmf.nist.gov/22.14.E10 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

22.14.11		$\int\operatorname{ds}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus A&S Ref: 16.24.11 Permalink: http://dlmf.nist.gov/22.14.E11 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

22.14.12		$\int\operatorname{cs}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{ds}\left(x,k\right)\right).$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus A&S Ref: 16.24.12 Permalink: http://dlmf.nist.gov/22.14.E12 Encodings: TeX, pMML, png See also: Annotations for §22.14(i), §22.14 and Ch.22

For alternative, and symmetric, formulations of the results in this subsection see Carlson (2006a).

§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions

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Permalink:: http://dlmf.nist.gov/22.14.ii
See also:: Annotations for §22.14 and Ch.22

See §22.16(ii). The indefinite integral of the 3rd power of a Jacobian function can be expressed as an elementary function of Jacobian functions and a product of Jacobian functions. The indefinite integral of a 4th power can be expressed as a complete elliptic integral, a polynomial in Jacobian functions, and the integration variable. See Lawden (1989, pp. 87–88). See also Gradshteyn and Ryzhik (2015, §§5.131–5.134) and Carlson (2006a).

For indefinite integrals of squares and products of even powers of Jacobian functions in terms of symmetric elliptic integrals, see Carlson (2006b).

§22.14(iii) Other Indefinite Integrals

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Keywords:: Jacobian elliptic functions, indefinite, integrals
Notes:: See Lawden (1989, pp. 40–41), Whittaker and Watson (1927, pp. 514–517).
Permalink:: http://dlmf.nist.gov/22.14.iii
See also:: Annotations for §22.14 and Ch.22

In (22.14.13)–(22.14.15), $0<x<2K$ .

22.14.13		$\int\frac{\,\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\ln\left(\frac{% \operatorname{sn}\left(x,k\right)}{\operatorname{cn}\left(x,k\right)+% \operatorname{dn}\left(x,k\right)}\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus Referenced by: §22.14(iii) Permalink: http://dlmf.nist.gov/22.14.E13 Encodings: TeX, pMML, png See also: Annotations for §22.14(iii), §22.14 and Ch.22

22.14.14		$\int\frac{\operatorname{cn}\left(x,k\right)\,\mathrm{d}x}{\operatorname{sn}% \left(x,k\right)}=\frac{1}{2}\ln\left(\frac{1-\operatorname{dn}\left(x,k\right% )}{1+\operatorname{dn}\left(x,k\right)}\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus Permalink: http://dlmf.nist.gov/22.14.E14 Encodings: TeX, pMML, png See also: Annotations for §22.14(iii), §22.14 and Ch.22

22.14.15		$\int\frac{\operatorname{cn}\left(x,k\right)\,\mathrm{d}x}{{\operatorname{sn}}^% {2}\left(x,k\right)}=-\frac{\operatorname{dn}\left(x,k\right)}{\operatorname{% sn}\left(x,k\right)}.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus Referenced by: §22.14(iii) Permalink: http://dlmf.nist.gov/22.14.E15 Encodings: TeX, pMML, png See also: Annotations for §22.14(iii), §22.14 and Ch.22

For additional results see Gradshteyn and Ryzhik (2015, §5.13) and Lawden (1989, Chapter 3).

§22.14(iv) Definite Integrals

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Keywords:: Jacobian elliptic functions, definite, integrals
Notes:: See Whittaker and Watson (1927, pp. 508–509).
Permalink:: http://dlmf.nist.gov/22.14.iv
See also:: Annotations for §22.14 and Ch.22

22.14.16		$\int_{0}^{K\left(k\right)}\ln\left(\operatorname{sn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)-\tfrac{1}{2}K\left(k% \right)\ln k,$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $k$ : modulus Referenced by: Erratum (V1.0.28) for Equations (22.14.16), (22.14.17) Permalink: http://dlmf.nist.gov/22.14.E16 Encodings: TeX, pMML, png Correction (effective with 1.0.28): Originally, a factor of $\pi$ was missing from the term containing the $\tfrac{1}{4}{K^{\prime}}\left(k\right)$ . Suggested 2020-08-06 by Fred Hucht See also: Annotations for §22.14(iv), §22.14 and Ch.22

22.14.17		$\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)+\tfrac{1}{2}K\left(k% \right)\ln\left(k^{\prime}/k\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : modulus and $k^{\prime}$ : complementary modulus Referenced by: Erratum (V1.0.28) for Equations (22.14.16), (22.14.17) Permalink: http://dlmf.nist.gov/22.14.E17 Encodings: TeX, pMML, png Correction (effective with 1.0.28): Originally, a factor of $\pi$ was missing from the term containing the $\tfrac{1}{4}{K^{\prime}}\left(k\right)$ . Suggested 2020-08-06 by Fred Hucht See also: Annotations for §22.14(iv), §22.14 and Ch.22

22.14.18		$\int_{0}^{K\left(k\right)}\ln\left(\operatorname{dn}\left(t,k\right)\right)\,% \mathrm{d}t=\tfrac{1}{2}K\left(k\right)\ln k^{\prime}.$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.14.E18 Encodings: TeX, pMML, png See also: Annotations for §22.14(iv), §22.14 and Ch.22

Corresponding results for the subsidiary functions follow by subtraction; compare (22.2.10).

22.13 Derivatives and Differential Equations 22.15 Inverse Functions

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§22.14 Integrals

Contents

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions

§22.14(iii) Other Indefinite Integrals

§22.14(iv) Definite Integrals