22 Jacobian Elliptic Functions Properties 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series 22.14 Integrals

§22.13 Derivatives and Differential Equations

Permalink:: http://dlmf.nist.gov/22.13

§22.13(i) Derivatives
§22.13(ii) First-Order Differential Equations
§22.13(iii) Second-Order Differential Equations

§22.13(i) Derivatives

Notes:: See Lawden (1989, §2.5), Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 491–494).
Keywords:: Jacobian elliptic functions, derivatives
Permalink:: http://dlmf.nist.gov/22.13.i

Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

$\frac{d}{dz}(\mathop{\mathrm{sn}\/}\nolimits z)$ $\;=\;$	$\mathop{\mathrm{cn}\/}\nolimits z\mathop{\mathrm{dn}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{dc}\/}\nolimits z)$ =	${k^{{\prime}}}^{2}\mathop{\mathrm{sc}\/}\nolimits z\mathop{\mathrm{nc}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{cn}\/}\nolimits z)$ $\;=\;$	$-\mathop{\mathrm{sn}\/}\nolimits z\mathop{\mathrm{dn}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{nc}\/}\nolimits z)$ =	$\mathop{\mathrm{sc}\/}\nolimits z\mathop{\mathrm{dc}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{dn}\/}\nolimits z)$ $\;=\;$	$-k^{2}\mathop{\mathrm{sn}\/}\nolimits z\mathop{\mathrm{cn}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{sc}\/}\nolimits z)$ =	$\mathop{\mathrm{dc}\/}\nolimits z\mathop{\mathrm{nc}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{cd}\/}\nolimits z)$ $\;=\;$	$-{k^{{\prime}}}^{2}\mathop{\mathrm{sd}\/}\nolimits z\mathop{\mathrm{nd}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{ns}\/}\nolimits z)$ =	$-\mathop{\mathrm{ds}\/}\nolimits z\mathop{\mathrm{cs}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{sd}\/}\nolimits z)$ $\;=\;$	$\mathop{\mathrm{cd}\/}\nolimits z\mathop{\mathrm{nd}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{ds}\/}\nolimits z)$ =	$-\mathop{\mathrm{cs}\/}\nolimits z\mathop{\mathrm{ns}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{nd}\/}\nolimits z)$ $\;=\;$	$k^{2}\mathop{\mathrm{sd}\/}\nolimits z\mathop{\mathrm{cd}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{cs}\/}\nolimits z)$ =	$-\mathop{\mathrm{ns}\/}\nolimits z\mathop{\mathrm{ds}\/}\nolimits z$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: Table 16.16
Referenced by:: §22.13(i)
Permalink:: http://dlmf.nist.gov/22.13.T1

Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. (The modulus is suppressed throughout the table.)

For alternative, and symmetric, formulations of these results see Carlson (2004, 2006a).

§22.13(ii) First-Order Differential Equations

Notes:: See Whittaker and Watson (1927, pp. 491–498).
Keywords:: Jacobian elliptic functions, differential equations
Permalink:: http://dlmf.nist.gov/22.13.ii

22.13.1 $\left(\frac{d}{dz}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)\right)^{2}=% \left(1-{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left(1-% k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right)\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Referenced by:: §22.13(iii), §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.13.E1
Encodings:: TeX, pMML, png

22.13.2 $\left(\frac{d}{dz}\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)\right)^{2}={% \left(1-{\mathop{\mathrm{cn}\/}\nolimits^{{2}}}\left(z,k\right)\right)}{\left(% {k^{{\prime}}}^{2}+k^{2}{\mathop{\mathrm{cn}\/}\nolimits^{{2}}}\left(z,k\right% )\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E2
Encodings:: TeX, pMML, png

22.13.3 $\left(\frac{d}{dz}\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)\right)^{2}=% \left(1-{\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left({% \mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z,k\right)-{k^{{\prime}}}^{2}% \right).$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E3
Encodings:: TeX, pMML, png

22.13.4 $\left(\frac{d}{dz}\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)\right)^{2}=% \left(1-{\mathop{\mathrm{cd}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left(1-% k^{2}{\mathop{\mathrm{cd}\/}\nolimits^{{2}}}\left(z,k\right)\right),$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.13.E4
Encodings:: TeX, pMML, png

22.13.5 $\left(\frac{d}{dz}\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)\right)^{2}={% \left(1-{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}\left(z,k% \right)\right)}{\left(1+k^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}\left(z,k% \right)\right)},$

Symbols:: $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E5
Encodings:: TeX, pMML, png

22.13.6 $\left(\frac{d}{dz}\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)\right)^{2}=% \left({\mathop{\mathrm{nd}\/}\nolimits^{{2}}}\left(z,k\right)-1\right)\left(1-% {k^{{\prime}}}^{2}{\mathop{\mathrm{nd}\/}\nolimits^{{2}}}\left(z,k\right)% \right),$

Symbols:: $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E6
Encodings:: TeX, pMML, png

22.13.7 $\left(\frac{d}{dz}\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)\right)^{2}=% \left({\mathop{\mathrm{dc}\/}\nolimits^{{2}}}\left(z,k\right)-1\right)\left({% \mathop{\mathrm{dc}\/}\nolimits^{{2}}}\left(z,k\right)-k^{2}\right),$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.13.E7
Encodings:: TeX, pMML, png

22.13.8 $\left(\frac{d}{dz}\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)\right)^{2}={% \left(k^{2}+{k^{{\prime}}}^{2}{\mathop{\mathrm{nc}\/}\nolimits^{{2}}}\left(z,k% \right)\right)}{\left({\mathop{\mathrm{nc}\/}\nolimits^{{2}}}\left(z,k\right)-% 1\right)},$

Symbols:: $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E8
Encodings:: TeX, pMML, png

22.13.9 $\left(\frac{d}{dz}\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)\right)^{2}=% \left(1+{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left(1+% {k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(z,k\right)% \right),$

Symbols:: $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E9
Encodings:: TeX, pMML, png

22.13.10 $\left(\frac{d}{dz}\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)\right)^{2}=% \left({\mathop{\mathrm{ns}\/}\nolimits^{{2}}}\left(z,k\right)-k^{2}\right)% \left({\mathop{\mathrm{ns}\/}\nolimits^{{2}}}\left(z,k\right)-1\right),$

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.13.E10
Encodings:: TeX, pMML, png

22.13.11 $\left(\frac{d}{dz}\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)\right)^{2}=% \left({\mathop{\mathrm{ds}\/}\nolimits^{{2}}}\left(z,k\right)-{k^{{\prime}}}^{% 2}\right)\left(k^{2}+{\mathop{\mathrm{ds}\/}\nolimits^{{2}}}\left(z,k\right)% \right),$

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E11
Encodings:: TeX, pMML, png

22.13.12 $\left(\frac{d}{dz}\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)\right)^{2}=% \left(1+{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left({k% ^{{\prime}}}^{2}+{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(z,k\right)\right).$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.13(iii)
Permalink:: http://dlmf.nist.gov/22.13.E12
Encodings:: TeX, pMML, png

For alternative, and symmetric, formulations of these results see Carlson (2006a).

§22.13(iii) Second-Order Differential Equations

Notes:: Equations (22.13.13)–(22.13.24) are obtained by differentiation of (22.13.1)–(22.13.12).
Keywords:: Jacobian elliptic functions, differential equations
Permalink:: http://dlmf.nist.gov/22.13.iii

22.13.13 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=-(1+k^% {2})\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)+2k^{2}{\mathop{\mathrm{sn}% \/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Referenced by:: §22.10(i), §22.10(ii), §22.13(iii)
Permalink:: http://dlmf.nist.gov/22.13.E13
Encodings:: TeX, pMML, png

22.13.14 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=-({k^{% {\prime}}}^{2}-k^{2})\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)-2k^{2}{% \mathop{\mathrm{cn}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.10(i), §22.10(ii)
Permalink:: http://dlmf.nist.gov/22.13.E14
Encodings:: TeX, pMML, png

22.13.15 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=(1+{k^% {{\prime}}}^{2})\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)-2{\mathop{% \mathrm{dn}\/}\nolimits^{{3}}}\left(z,k\right).$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.10(i), §22.10(ii)
Permalink:: http://dlmf.nist.gov/22.13.E15
Encodings:: TeX, pMML, png

22.13.16 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)=-(1+k^% {2})\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)+2k^{2}{\mathop{\mathrm{cd}% \/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.13.E16
Encodings:: TeX, pMML, png

22.13.17 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)=(k^{2}% -{k^{{\prime}}}^{2})\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)-2k^{2}{k^{% {\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E17
Encodings:: TeX, pMML, png

22.13.18 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)=(1+{k^% {{\prime}}}^{2})\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)-2{k^{{\prime}}% }^{2}{\mathop{\mathrm{nd}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E18
Encodings:: TeX, pMML, png

22.13.19 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)=-(1+k^% {2})\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)+2{\mathop{\mathrm{dc}\/}% \nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.13.E19
Encodings:: TeX, pMML, png

22.13.20 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)=(k^{2}% -{k^{{\prime}}}^{2})\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)+2{k^{{% \prime}}}^{2}{\mathop{\mathrm{nc}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E20
Encodings:: TeX, pMML, png

22.13.21 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)=(1+{k^% {{\prime}}}^{2})\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)+2{k^{{\prime}}% }^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E21
Encodings:: TeX, pMML, png

22.13.22 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)=-(1+k^% {2})\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)+2{\mathop{\mathrm{ns}\/}% \nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.13.E22
Encodings:: TeX, pMML, png

22.13.23 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)=(k^{2}% -{k^{{\prime}}}^{2})\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)+2{\mathop{% \mathrm{ds}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E23
Encodings:: TeX, pMML, png

22.13.24 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)=(1+{k^% {{\prime}}}^{2})\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)+2{\mathop{% \mathrm{cs}\/}\nolimits^{{3}}}\left(z,k\right).$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex, : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.13(iii)
Permalink:: http://dlmf.nist.gov/22.13.E24
Encodings:: TeX, pMML, png

For alternative, and symmetric, formulations of these results see Carlson (2006a).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series 22.14 Integrals

§22.13 Derivatives and Differential Equations

Contents

§22.13(i) Derivatives

§22.13(ii) First-Order Differential Equations

§22.13(iii) Second-Order Differential Equations