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

§19.4 Derivatives and Differential Equations

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

§19.4(i) Derivatives

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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	$\displaystyle\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}$	$\displaystyle=\frac{E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)}{k{k^{% \prime}}^{2}},$
19.4.1	$\displaystyle\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))% }{\mathrm{d}k}$	$\displaystyle=kK\left(k\right),$
ⓘ 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 $f$ with respect to $x$ , $k$ : 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	$\displaystyle\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}$	$\displaystyle=\frac{E\left(k\right)-K\left(k\right)}{k},$
19.4.2	$\displaystyle\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}$	$\displaystyle=-\frac{kE\left(k\right)}{{k^{\prime}}^{2}},$
ⓘ 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 $f$ with respect to $x$ , $k$ : 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 $f$ with respect to $x$ , $k$ : 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 $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $k$ : 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 $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : 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 $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument and $k$ : 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 $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : 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

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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 $k$ . 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, $k$ : 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, $k$ : 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

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§19.4 Derivatives and Differential Equations

Contents

§19.4(i) Derivatives

§19.4(ii) Differential Equations