§22.14 Integrals

Permalink:: http://dlmf.nist.gov/22.14

§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

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

With $x\in\Real$ ,

22.14.1 $\int\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)dx=k^{{-1}}\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)-k\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

22.14.2 $\int\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)dx=k^{{-1}}\mathop{\mathrm{Arccos}\/}\nolimits\!\left(\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{Arccos}\/}\nolimits z$ : 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

22.14.3 $\int\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)dx=\mathop{\mathrm{Arcsin}\/}\nolimits\!\left(\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)\right)=\mathop{\mathrm{am}\/}\nolimits\left(x,k\right).$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : general arcsine function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.3
Permalink:: http://dlmf.nist.gov/22.14.E3
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The branches of the inverse trigonometric functions are chosen so that they are continuous. See §22.16(i) for $\mathop{\mathrm{am}\/}\nolimits\left(z,k\right)$ .

Secondly,

22.14.4 $\int\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right)dx=k^{{-1}}\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{nd}\/}\nolimits\left(x,k\right)+k\mathop{\mathrm{sd}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

22.14.5 $\int\mathop{\mathrm{sd}\/}\nolimits\left(x,k\right)dx=(kk^{{\prime}})^{{-1}}\mathop{\mathrm{Arcsin}\/}\nolimits\!\left(-k\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : 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

22.14.6 $\int\mathop{\mathrm{nd}\/}\nolimits\left(x,k\right)dx={k^{{\prime}}}^{{-1}}\mathop{\mathrm{Arccos}\/}\nolimits\!\left(\mathop{\mathrm{cd}\/}\nolimits\left(x,k\right)\right).$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{Arccos}\/}\nolimits z$ : 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

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

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

22.14.7 $\int\mathop{\mathrm{dc}\/}\nolimits\left(x,k\right)dx=\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{nc}\/}\nolimits\left(x,k\right)+\mathop{\mathrm{sc}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

22.14.8 $\int\mathop{\mathrm{nc}\/}\nolimits\left(x,k\right)dx={k^{{\prime}}}^{{-1}}\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{dc}\/}\nolimits\left(x,k\right)+k^{{\prime}}\mathop{\mathrm{sc}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

22.14.9 $\int\mathop{\mathrm{sc}\/}\nolimits\left(x,k\right)dx={k^{{\prime}}}^{{-1}}\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{dc}\/}\nolimits\left(x,k\right)+k^{{\prime}}\mathop{\mathrm{nc}\/}\nolimits\left(x,k\right)\right).$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

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

22.14.10 $\int\mathop{\mathrm{ns}\/}\nolimits\left(x,k\right)dx=\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{ds}\/}\nolimits\left(x,k\right)-\mathop{\mathrm{cs}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

22.14.11 $\int\mathop{\mathrm{ds}\/}\nolimits\left(x,k\right)dx=\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{ns}\/}\nolimits\left(x,k\right)-\mathop{\mathrm{cs}\/}\nolimits\left(x,k\right)\right),$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

22.14.12 $\int\mathop{\mathrm{cs}\/}\nolimits\left(x,k\right)dx=\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{ns}\/}\nolimits\left(x,k\right)-\mathop{\mathrm{ds}\/}\nolimits\left(x,k\right)\right).$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

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

Permalink:: http://dlmf.nist.gov/22.14.ii

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 (2000, pp. 618–619) 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

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

In (22.14.13)–(22.14.15), $0<x<2K$ .

22.14.13 $\int\frac{dx}{\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)}=\mathop{\ln\/}\nolimits\!\left(\frac{\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)}{\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)+\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)}\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

22.14.14 $\int\frac{\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)dx}{\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)}=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{1-\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)}{1+\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)}\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.14.E14
Encodings:: TeX, pMML, png

22.14.15 $\int\frac{\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)dx}{{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(x,k\right)}=-\frac{\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)}{\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)}.$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $dx$ : 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

For additional results see Gradshteyn and Ryzhik (2000, pp. 619–622) and Lawden (1989, Chapter 3).

§22.14(iv) Definite Integrals

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

22.14.16 $\int _{0}^{{\mathop{K\/}\nolimits\!\left(k\right)}}\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{sn}\/}\nolimits\left(t,k\right)\right)dt=-\tfrac{1}{4}\!\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)-\tfrac{1}{2}\!\mathop{K\/}\nolimits\!\left(k\right)\mathop{\ln\/}\nolimits k,$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.14.E16
Encodings:: TeX, pMML, png

22.14.17 $\int _{0}^{{\mathop{K\/}\nolimits\!\left(k\right)}}\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{cn}\/}\nolimits\left(t,k\right)\right)dt=-\tfrac{1}{4}\!\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)+\tfrac{1}{2}\!\mathop{K\/}\nolimits\!\left(k\right)\mathop{\ln\/}\nolimits\!\left(k^{{\prime}}/k\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.14.E17
Encodings:: TeX, pMML, png

22.14.18 $\int _{0}^{{\mathop{K\/}\nolimits\!\left(k\right)}}\mathop{\ln\/}\nolimits\!\left(\mathop{\mathrm{dn}\/}\nolimits\left(t,k\right)\right)dt=\tfrac{1}{2}\!\mathop{K\/}\nolimits\!\left(k\right)\mathop{\ln\/}\nolimits k^{{\prime}}.$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.14.E18
Encodings:: TeX, pMML, png

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

§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