19 Elliptic Integrals Legendre’s Integrals 19.3 Graphics 19.5 Maclaurin and Related Expansions

§19.4 Derivatives and Differential Equations

ⓘ

Permalink:: http://dlmf.nist.gov/19.4
See also:: Annotations for Ch.19

§19.4(i) Derivatives
§19.4(ii) Differential Equations

§19.4(i) Derivatives

ⓘ

Keywords:: Legendre’s elliptic integrals, derivatives
Notes:: These results follow by differentiation of the definitions in §19.2(ii).
Permalink:: http://dlmf.nist.gov/19.4.i
See also:: Annotations for §19.4 and Ch.19

19.4.1

$\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-{k^{\prime% }}^{2}K\left(k\right)}{k{k^{\prime}}^{2}},$

$\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}{\mathrm{d}k% }=kK\left(k\right),$

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E1
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.4(i), §19.4 and Ch.19

19.4.2

$\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-K\left(k% \right)}{k},$

$\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}=-\frac{kE\left% (k\right)}{{k^{\prime}}^{2}},$

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.4(i), §19.4 and Ch.19

19.4.3 $\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{% \mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-% E\left(k\right)}{k^{2}{k^{\prime}}^{2}},$

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §19.4(i), §19.4 and Ch.19

19.4.4 $\frac{\partial\Pi\left(\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^% {2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{\prime}}^{2}\Pi\left(\alpha^{2},k% \right)).$

ⓘ

Symbols:: $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.4.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §19.4(i), §19.4 and Ch.19

19.4.5 $\frac{\partial F\left(\phi,k\right)}{\partial k}={\frac{E\left(\phi,k\right)-{% k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}^{2}}-\frac{k\sin\phi\cos% \phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin}^{2}\phi}}},$

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §19.4(i), §19.4 and Ch.19

19.4.6 $\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F% \left(\phi,k\right)}{k},$

ⓘ

Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , $\phi$ : real or complex argument and : real or complex modulus
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.4.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §19.4(i), §19.4 and Ch.19

19.4.7 $\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{% \prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k\right)-{k^{\prime}}^{2}\Pi% \left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi\cos\phi}{\sqrt{1-k^{2}{% \sin}^{2}\phi}}\right).$

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.4.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §19.4(i), §19.4 and Ch.19

§19.4(ii) Differential Equations

ⓘ

Keywords:: Legendre’s elliptic integrals, differential equations
Notes:: See Cazenave (1969, p. 175). (19.4.8) agrees also with Edwards (1954, vol. 1, p. 402) and with expansion to first order in . The term on the right side in Byrd and Friedman (1971, 118.01) has the wrong sign.
Permalink:: http://dlmf.nist.gov/19.4.ii
See also:: Annotations for §19.4 and Ch.19

Let $D_{k}=\ifrac{\partial}{\partial k}$ . Then

19.4.8 $(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k% \sin\phi\cos\phi}{(1-k^{2}{\sin}^{2}\phi)^{3/2}},$

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, : real or complex modulus, $k^{\prime}$ : complementary modulus and $D_{k}$ : differential operator
Referenced by:: §19.4(ii)
Permalink:: http://dlmf.nist.gov/19.4.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §19.4(ii), §19.4 and Ch.19

19.4.9 $(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac% {k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin}^{2}\phi}}.$

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, : real or complex modulus, $k^{\prime}$ : complementary modulus and $D_{k}$ : differential operator
Permalink:: http://dlmf.nist.gov/19.4.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §19.4(ii), §19.4 and Ch.19

If $\phi=\pi/2$ , then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$ . An analogous differential equation of third order for $\Pi\left(\phi,\alpha^{2},k\right)$ is given in Byrd and Friedman (1971, 118.03).

19.3 Graphics 19.5 Maclaurin and Related Expansions

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

§19.4 Derivatives and Differential Equations

Contents

§19.4(i) Derivatives

§19.4(ii) Differential Equations