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
See also:: Annotations for Ch.22

§22.13(i) Derivatives

ⓘ

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

ⓘ

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

ⓘ

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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§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

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.