§19.4 Derivatives and Differential Equations

Permalink:: http://dlmf.nist.gov/19.4

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

§19.4(i) Derivatives

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

19.4.1

$\frac{d\mathop{K\/}\nolimits\!\left(k\right)}{dk}=\frac{\mathop{E\/}\nolimits% \!\left(k\right)-{k^{{\prime}}}^{2}\mathop{K\/}\nolimits\!\left(k\right)}{k{k^% {{\prime}}}^{2}},$

$\frac{d(\mathop{E\/}\nolimits\!\left(k\right)-{k^{{\prime}}}^{2}\mathop{K\/}% \nolimits\!\left(k\right))}{dk}=k\mathop{K\/}\nolimits\!\left(k\right),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$ : derivative of with respect to , : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E1
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19.4.2

$\frac{d\mathop{E\/}\nolimits\!\left(k\right)}{dk}=\frac{\mathop{E\/}\nolimits% \!\left(k\right)-\mathop{K\/}\nolimits\!\left(k\right)}{k},$

$\frac{d(\mathop{E\/}\nolimits\!\left(k\right)-\mathop{K\/}\nolimits\!\left(k% \right))}{dk}=-\frac{k\mathop{E\/}\nolimits\!\left(k\right)}{{k^{{\prime}}}^{2% }},$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$ : derivative of with respect to , : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E2
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19.4.3 $\frac{{d}^{2}\mathop{E\/}\nolimits\!\left(k\right)}{{dk}^{2}}=-\frac{1}{k}% \frac{d\mathop{K\/}\nolimits\!\left(k\right)}{dk}=\frac{{k^{{\prime}}}^{2}% \mathop{K\/}\nolimits\!\left(k\right)-\mathop{E\/}\nolimits\!\left(k\right)}{k% ^{2}{k^{{\prime}}}^{2}},$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$ : derivative of with respect to , : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E3
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19.4.4 $\frac{\partial\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)}{\partial k}=% \frac{k}{{k^{{\prime}}}^{2}(k^{2}-\alpha^{2})}(\mathop{E\/}\nolimits\!\left(k% \right)-{k^{{\prime}}}^{2}\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)).$

Symbols:: $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : 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

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E5
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19.4.6 $\frac{\partial\mathop{E\/}\nolimits\!\left(\phi,k\right)}{\partial k}=\frac{% \mathop{E\/}\nolimits\!\left(\phi,k\right)-\mathop{F\/}\nolimits\!\left(\phi,k% \right)}{k},$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\phi$ : real or complex argument and : real or complex modulus
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.4.E6
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19.4.7 $\frac{\partial\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)}{% \partial k}=\frac{k}{{k^{{\prime}}}^{2}(k^{2}-\alpha^{2})}\left({\mathop{E\/}% \nolimits\!\left(\phi,k\right)-{k^{{\prime}}}^{2}\mathop{\Pi\/}\nolimits\!% \left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\mathop{\sin\/}\nolimits\phi\mathop% {\cos\/}\nolimits\phi}{\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi}}% \right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : 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
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§19.4(ii) 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.
Keywords:: Legendre’s elliptic integrals, differential equations
Permalink:: http://dlmf.nist.gov/19.4.ii

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

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : 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
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19.4.9 $(k{k^{{\prime}}}^{2}D_{k}^{2}+{k^{{\prime}}}^{2}D_{k}+k)\mathop{E\/}\nolimits% \!\left(\phi,k\right)=\frac{k\mathop{\sin\/}\nolimits\phi\mathop{\cos\/}% \nolimits\phi}{\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : 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
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If $\phi=\pi/2$ , then these two equations become hypergeometric differential equations (15.10.1) for $\mathop{K\/}\nolimits\!\left(k\right)$ and $\mathop{E\/}\nolimits\!\left(k\right)$ . An analogous differential equation of third order for $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ is given in Byrd and Friedman (1971, 118.03).

§19.4 Derivatives and Differential Equations

Contents

§19.4(i) Derivatives

§19.4(ii) Differential Equations