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

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Permalink:: http://dlmf.nist.gov/22.13
See also:: Annotations for Ch.22

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

§22.13(i) Derivatives

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Keywords:: Jacobian elliptic functions, derivatives
Notes:: See Lawden (1989, §2.5), Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 491–494).
Permalink:: http://dlmf.nist.gov/22.13.i
See also:: Annotations for §22.13 and Ch.22

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

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

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

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

§22.13(ii) First-Order Differential Equations

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Keywords:: Jacobian elliptic functions, differential equations, first-order
Notes:: See Whittaker and Watson (1927, pp. 491–498).
Permalink:: http://dlmf.nist.gov/22.13.ii
See also:: Annotations for §22.13 and Ch.22

22.13.1	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left(1-{\operatorname{sn}}^{2}\left(z,k\right)\right)\left(1-k^% {2}{\operatorname{sn}}^{2}\left(z,k\right)\right),$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Referenced by: §22.13(iii), §22.18(iv) Permalink: http://dlmf.nist.gov/22.13.E1 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.2	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k% \right)\right)^{2}$	$\displaystyle={\left(1-{\operatorname{cn}}^{2}\left(z,k\right)\right)}{\left({% k^{\prime}}^{2}+k^{2}{\operatorname{cn}}^{2}\left(z,k\right)\right)},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E2 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.3	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dn}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left(1-{\operatorname{dn}}^{2}\left(z,k\right)\right)\left({% \operatorname{dn}}^{2}\left(z,k\right)-{k^{\prime}}^{2}\right).$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E3 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22

22.13.4	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cd}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left(1-{\operatorname{cd}}^{2}\left(z,k\right)\right)\left(1-k^% {2}{\operatorname{cd}}^{2}\left(z,k\right)\right),$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.13.E4 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.5	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sd}\left(z,k% \right)\right)^{2}$	$\displaystyle={\left(1-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right)% \right)}{\left(1+k^{2}{\operatorname{sd}}^{2}\left(z,k\right)\right)},$
ⓘ Symbols: $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E5 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.6	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nd}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left({\operatorname{nd}}^{2}\left(z,k\right)-1\right)\left(1-{k% ^{\prime}}^{2}{\operatorname{nd}}^{2}\left(z,k\right)\right),$
ⓘ Symbols: $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E6 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.7	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left({\operatorname{dc}}^{2}\left(z,k\right)-1\right)\left({% \operatorname{dc}}^{2}\left(z,k\right)-k^{2}\right),$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.13.E7 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.8	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nc}\left(z,k% \right)\right)^{2}$	$\displaystyle={\left(k^{2}+{k^{\prime}}^{2}{\operatorname{nc}}^{2}\left(z,k% \right)\right)}{\left({\operatorname{nc}}^{2}\left(z,k\right)-1\right)},$
ⓘ Symbols: $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E8 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.9	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sc}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left(1+{\operatorname{sc}}^{2}\left(z,k\right)\right)\left(1+{k% ^{\prime}}^{2}{\operatorname{sc}}^{2}\left(z,k\right)\right),$
ⓘ Symbols: $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E9 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.10	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ns}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left({\operatorname{ns}}^{2}\left(z,k\right)-k^{2}\right)\left(% {\operatorname{ns}}^{2}\left(z,k\right)-1\right),$
ⓘ Symbols: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.13.E10 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.11	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ds}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left({\operatorname{ds}}^{2}\left(z,k\right)-{k^{\prime}}^{2}% \right)\left(k^{2}+{\operatorname{ds}}^{2}\left(z,k\right)\right),$
ⓘ Symbols: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E11 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22
22.13.12	$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cs}\left(z,k% \right)\right)^{2}$	$\displaystyle=\left(1+{\operatorname{cs}}^{2}\left(z,k\right)\right)\left({k^{% \prime}}^{2}+{\operatorname{cs}}^{2}\left(z,k\right)\right).$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Referenced by: §22.13(iii) Permalink: http://dlmf.nist.gov/22.13.E12 Encodings: TeX, pMML, png See also: Annotations for §22.13(ii), §22.13 and Ch.22

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

§22.13(iii) Second-Order Differential Equations

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Keywords:: Jacobian elliptic functions, differential equations, second-order
Notes:: Equations (22.13.13)–(22.13.24) are obtained by differentiation of (22.13.1)–(22.13.12).
Permalink:: http://dlmf.nist.gov/22.13.iii
See also:: Annotations for §22.13 and Ch.22

22.13.13	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sn}\left(% z,k\right)$	$\displaystyle=-(1+k^{2})\operatorname{sn}\left(z,k\right)+2k^{2}{\operatorname% {sn}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Referenced by: §22.10(i), §22.10(ii), §22.13(iii) Permalink: http://dlmf.nist.gov/22.13.E13 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.14	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cn}\left(% z,k\right)$	$\displaystyle=-({k^{\prime}}^{2}-k^{2})\operatorname{cn}\left(z,k\right)-2k^{2% }{\operatorname{cn}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : 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 See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.15	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dn}\left(% z,k\right)$	$\displaystyle=(1+{k^{\prime}}^{2})\operatorname{dn}\left(z,k\right)-2{% \operatorname{dn}}^{3}\left(z,k\right).$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : 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 See also: Annotations for §22.13(iii), §22.13 and Ch.22

22.13.16	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cd}\left(% z,k\right)$	$\displaystyle=-(1+k^{2})\operatorname{cd}\left(z,k\right)+2k^{2}{\operatorname% {cd}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.13.E16 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.17	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sd}\left(% z,k\right)$	$\displaystyle=(k^{2}-{k^{\prime}}^{2})\operatorname{sd}\left(z,k\right)-2k^{2}% {k^{\prime}}^{2}{\operatorname{sd}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E17 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.18	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{nd}\left(% z,k\right)$	$\displaystyle=(1+{k^{\prime}}^{2})\operatorname{nd}\left(z,k\right)-2{k^{% \prime}}^{2}{\operatorname{nd}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E18 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.19	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dc}\left(% z,k\right)$	$\displaystyle=-(1+k^{2})\operatorname{dc}\left(z,k\right)+2{\operatorname{dc}}% ^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.13.E19 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.20	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{nc}\left(% z,k\right)$	$\displaystyle=(k^{2}-{k^{\prime}}^{2})\operatorname{nc}\left(z,k\right)+2{k^{% \prime}}^{2}{\operatorname{nc}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E20 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.21	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sc}\left(% z,k\right)$	$\displaystyle=(1+{k^{\prime}}^{2})\operatorname{sc}\left(z,k\right)+2{k^{% \prime}}^{2}{\operatorname{sc}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E21 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.22	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ns}\left(% z,k\right)$	$\displaystyle=-(1+k^{2})\operatorname{ns}\left(z,k\right)+2{\operatorname{ns}}% ^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.13.E22 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.23	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ds}\left(% z,k\right)$	$\displaystyle=(k^{2}-{k^{\prime}}^{2})\operatorname{ds}\left(z,k\right)+2{% \operatorname{ds}}^{3}\left(z,k\right),$
ⓘ Symbols: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.13.E23 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22
22.13.24	$\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cs}\left(% z,k\right)$	$\displaystyle=(1+{k^{\prime}}^{2})\operatorname{cs}\left(z,k\right)+2{% \operatorname{cs}}^{3}\left(z,k\right).$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Referenced by: §22.13(iii) Permalink: http://dlmf.nist.gov/22.13.E24 Encodings: TeX, pMML, png See also: Annotations for §22.13(iii), §22.13 and Ch.22

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

22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series 22.14 Integrals

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§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