19 Elliptic Integrals Symmetric Integrals 19.17 Graphics 19.19 Taylor and Related Series

§19.18 Derivatives and Differential Equations

Permalink:: http://dlmf.nist.gov/19.18

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

§19.18(i) Derivatives

Notes:: (19.18.1) is derived from (19.16.1), (19.16.5), and (19.18.4). (19.18.2) follows from (19.18.8). For (19.18.4) and (19.18.5) put and $c=a+a^{{\prime}}$ in Carlson (1977b, (5.9-9),(5.9-10)).
Keywords:: derivatives, hypergeometric -function, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.18.i

19.18.1 $\frac{\partial\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)}{\partial z}=-% \tfrac{1}{6}\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right),$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\frac{\partial f}{\partial x}$ : partial derivative of with respect to
Referenced by:: §19.18(i), §19.28
Permalink:: http://dlmf.nist.gov/19.18.E1
Encodings:: TeX, pMML, png

19.18.2 $\frac{d}{dx}\mathop{R_{G}\/}\nolimits\!\left(x+a,x+b,x+c\right)=\tfrac{1}{2}% \mathop{R_{F}\/}\nolimits\!\left(x+a,x+b,x+c\right).$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind and $\frac{df}{dx}$ : derivative of with respect to
Referenced by:: §19.18(i)
Permalink:: http://dlmf.nist.gov/19.18.E2
Encodings:: TeX, pMML, png

Let $\partial_{j}=\ifrac{\partial}{\partial z_{j}}$ , and $\mathbf{e}_{j}$ be an -tuple with 1 in the th place and 0’s elsewhere. Also define

19.18.3

$w_{j}=b_{j}\biggm/\sum_{{j=1}}^{n}b_{j},$

$a^{{\prime}}=-a+\sum_{{j=1}}^{n}b_{j}.$

Symbols:: : nonnegative integer and $w_{j}$
Permalink:: http://dlmf.nist.gov/19.18.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

The next two equations apply to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23).

19.18.4 $\partial_{j}\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=-% aw_{j}\mathop{R_{{-a-1}}\/}\nolimits\!\left(\mathbf{b}+\mathbf{e}_{j};\mathbf{% z}\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\partial x$ : partial differential of and $w_{j}$
Referenced by:: §19.18(i), §19.25(i)
Permalink:: http://dlmf.nist.gov/19.18.E4
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19.18.5 $(z_{j}\partial_{j}+b_{j})\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};% \mathbf{z}\right)=w_{j}a^{{\prime}}\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf% {b}+\mathbf{e}_{j};\mathbf{z}\right).$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\partial x$ : partial differential of and $w_{j}$
Referenced by:: §19.18(i)
Permalink:: http://dlmf.nist.gov/19.18.E5
Encodings:: TeX, pMML, png

§19.18(ii) Differential Equations

Notes:: (19.18.6) comes from (19.18.8) and (19.20.25). For (19.18.8) and (19.18.11) see Carlson (1977b, (5.9-2)). For (19.18.12)–(19.18.17) see Carlson (1977b, §5.4).
Keywords:: Euler–Poisson–Darboux equation, Euler–Poisson differential equations, Euler’s homogeneity relation, Laplace’s equation, axially symmetric potential theory, differential equations, hypergeometric -function, potential theory, symmetric elliptic integrals, wave equation
Permalink:: http://dlmf.nist.gov/19.18.ii

19.18.6 $\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\frac{-1}{2% \sqrt{xyz}},$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\frac{\partial f}{\partial x}$ : partial derivative of with respect to
Referenced by:: §19.18(ii), §19.32
Permalink:: http://dlmf.nist.gov/19.18.E6
Encodings:: TeX, pMML, png

19.18.7 $\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\tfrac{1}{2}% \mathop{R_{F}\/}\nolimits\!\left(x,y,z\right).$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind and $\frac{\partial f}{\partial x}$ : partial derivative of with respect to
Permalink:: http://dlmf.nist.gov/19.18.E7
Encodings:: TeX, pMML, png

19.18.8 $\sum_{{j=1}}^{n}\partial_{j}\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};% \mathbf{z}\right)=-a\mathop{R_{{-a-1}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z% }\right).$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\partial x$ : partial differential of and : nonnegative integer
Referenced by:: §19.18(i), §19.18(ii)
Permalink:: http://dlmf.nist.gov/19.18.E8
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19.18.9 $\left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{% \partial}{\partial z}\right)\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=-% \tfrac{1}{2}\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\frac{\partial f}{\partial x}$ : partial derivative of with respect to
Permalink:: http://dlmf.nist.gov/19.18.E9
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19.18.10 $\left((x-y)\frac{{\partial}^{2}}{\partial x\partial y}+\frac{1}{2}\left(\frac{% \partial}{\partial y}-\frac{\partial}{\partial x}\right)\right)\mathop{R_{F}\/% }\nolimits\!\left(x,y,z\right)=0,$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to and $\partial x$ : partial differential of
Referenced by:: §19.18(ii)
Permalink:: http://dlmf.nist.gov/19.18.E10
Encodings:: TeX, pMML, png

and two similar equations obtained by permuting in (19.18.10).

More concisely, if $v=\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ , then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation:

19.18.11 $\sum_{{j=1}}^{n}z_{j}\partial_{j}v=-av,$

Symbols:: $\partial x$ : partial differential of , : nonnegative integer and
Referenced by:: §19.18(ii), §19.18(ii)
Permalink:: http://dlmf.nist.gov/19.18.E11
Encodings:: TeX, pMML, png

and also a system of Euler–Poisson differential equations (of which only are independent):

19.18.12 $(z_{j}\partial_{j}+b_{j})\partial_{l}v=(z_{l}\partial_{l}+b_{l})\partial_{j}v,$

Symbols:: $\partial x$ : partial differential of , : nonnegative integer and
Referenced by:: §19.18(ii), §19.18(ii)
Permalink:: http://dlmf.nist.gov/19.18.E12
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or equivalently,

19.18.13 $((z_{j}-z_{l})\partial_{j}\partial_{l}+b_{j}\partial_{l}-b_{l}\partial_{j})v=0.$

Symbols:: $\partial x$ : partial differential of , : nonnegative integer and
Permalink:: http://dlmf.nist.gov/19.18.E13
Encodings:: TeX, pMML, png

Here $j,l=1,2,\dots,n$ and $j\neq l$ . For group-theoretical aspects of this system see Carlson (1963, §VI). If , then elimination of $\partial_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_{1},b_{2},z_{1},z_{2})=(b,c-b,1-z,1)$ , produces the Gauss hypergeometric equation (15.10.1).

The next four differential equations apply to the complete case of $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{G}\/}\nolimits$ in the form $\mathop{R_{{-a}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)$ (see (19.16.20) and (19.16.23)).

The function $w=\mathop{R_{{-a}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};x+y,x-y\right)$ satisfies an Euler–Poisson–Darboux equation:

19.18.14 $\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial w}{\partial y}.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to
Permalink:: http://dlmf.nist.gov/19.18.E14
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Also $W=\mathop{R_{{-a}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};t+r,t-r\right)$ , with $r=\sqrt{x^{2}+y^{2}}$ , satisfies a wave equation:

19.18.15 $\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{% 2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to and : wave function
Permalink:: http://dlmf.nist.gov/19.18.E15
Encodings:: TeX, pMML, png

Similarly, the function $u=\mathop{R_{{-a}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};x+iy,x-iy\right)$ satisfies an equation of axially symmetric potential theory:

19.18.16 $\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0,$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to
Permalink:: http://dlmf.nist.gov/19.18.E16
Encodings:: TeX, pMML, png

and $U=\mathop{R_{{-a}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};z+i\rho,z-i% \rho\right)$ , with $\rho=\sqrt{x^{2}+y^{2}}$ , satisfies Laplace’s equation:

19.18.17 $\frac{{\partial}^{2}U}{{\partial x}^{2}}+\frac{{\partial}^{2}U}{{\partial y}^{% 2}}+\frac{{\partial}^{2}U}{{\partial z}^{2}}=0.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to and
Referenced by:: §19.18(ii)
Permalink:: http://dlmf.nist.gov/19.18.E17
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.17 Graphics 19.19 Taylor and Related Series

§19.18 Derivatives and Differential Equations

Contents

§19.18(i) Derivatives

§19.18(ii) Differential Equations