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

§19.18 Derivatives and Differential Equations

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Permalink:: http://dlmf.nist.gov/19.18
See also:: Annotations for Ch.19

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

§19.18(i) Derivatives

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Keywords:: derivative, derivatives, hypergeometric $R$ -function, symmetric elliptic integrals
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 $t=-a$ and $c=a+a^{\prime}$ in Carlson (1977b, (5.9-9), (5.9-10)).
Permalink:: http://dlmf.nist.gov/19.18.i
See also:: Annotations for §19.18 and Ch.19

19.18.1		$\frac{\partial R_{F}\left(x,y,z\right)}{\partial z}=-\tfrac{1}{6}R_{D}\left(x,% y,z\right),$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$ Referenced by: §19.18(i), §19.28 Permalink: http://dlmf.nist.gov/19.18.E1 Encodings: TeX, pMML, png See also: Annotations for §19.18(i), §19.18 and Ch.19

19.18.2		$\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x+c\right)=\tfrac{1}{2}R_{F}% \left(x+a,x+b,x+c\right).$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ Referenced by: §19.18(i) Permalink: http://dlmf.nist.gov/19.18.E2 Encodings: TeX, pMML, png See also: Annotations for §19.18(i), §19.18 and Ch.19

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

19.18.3	$\displaystyle w_{j}$	$\displaystyle=b_{j}\biggm/\sum_{j=1}^{n}b_{j},$
19.18.3	$\displaystyle a^{\prime}$	$\displaystyle=-a+\sum_{j=1}^{n}b_{j}.$
ⓘ Symbols: $n$ : nonnegative integer and $w_{j}$ Permalink: http://dlmf.nist.gov/19.18.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.18(i), §19.18 and Ch.19

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

19.18.4		$\partial_{j}R_{-a}\left(\mathbf{b};\mathbf{z}\right)=-aw_{j}R_{-a-1}\left(% \mathbf{b}+\mathbf{e}_{j};\mathbf{z}\right),$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\,\partial\NVar{x}$ : partial differential of $x$ and $w_{j}$ Referenced by: §19.18(i), §19.25(i) Permalink: http://dlmf.nist.gov/19.18.E4 Encodings: TeX, pMML, png See also: Annotations for §19.18(i), §19.18 and Ch.19

19.18.5		$(z_{j}\partial_{j}+b_{j})R_{-a}\left(\mathbf{b};\mathbf{z}\right)=w_{j}a^{% \prime}R_{-a}\left(\mathbf{b}+\mathbf{e}_{j};\mathbf{z}\right).$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\,\partial\NVar{x}$ : partial differential of $x$ and $w_{j}$ Referenced by: §19.18(i) Permalink: http://dlmf.nist.gov/19.18.E5 Encodings: TeX, pMML, png See also: Annotations for §19.18(i), §19.18 and Ch.19

§19.18(ii) Differential Equations

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Keywords:: Euler–Poisson differential equations, Euler–Poisson–Darboux equation, Euler’s homogeneity relation, Laplace’s equation, axially symmetric potential theory, differential equation, differential equations, hypergeometric $R$ -function, potential theory, symmetric elliptic integrals, wave equation
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).
Permalink:: http://dlmf.nist.gov/19.18.ii
See also:: Annotations for §19.18 and Ch.19

19.18.6		$\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)R_{F}\left(x,y,z\right)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$ Referenced by: §19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.18.E6 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior. Suggested 2021-06-07 by Luc Maisonobe See also: Annotations for §19.18(ii), §19.18 and Ch.19

19.18.7		$\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)R_{G}\left(x,y,z\right)=\tfrac{1}{2}R_{F}\left(x,y,z\right).$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$ Referenced by: §19.25(vi) Permalink: http://dlmf.nist.gov/19.18.E7 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

19.18.8		$\sum_{j=1}^{n}\partial_{j}R_{-a}\left(\mathbf{b};\mathbf{z}\right)=-aR_{-a-1}% \left(\mathbf{b};\mathbf{z}\right).$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\,\partial\NVar{x}$ : partial differential of $x$ and $n$ : nonnegative integer Referenced by: §19.18(i), §19.18(ii) Permalink: http://dlmf.nist.gov/19.18.E8 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

19.18.9		$\left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{% \partial}{\partial z}\right)R_{F}\left(x,y,z\right)=-\tfrac{1}{2}R_{F}\left(x,% y,z\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$ Permalink: http://dlmf.nist.gov/19.18.E9 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

19.18.10		$\left((x-y)\frac{\,{\partial}^{2}}{\,\partial x\,\partial y}+\frac{1}{2}\left(% \frac{\partial}{\partial y}-\frac{\partial}{\partial x}\right)\right)R_{F}% \left(x,y,z\right)=0,$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$ Referenced by: §19.18(ii) Permalink: http://dlmf.nist.gov/19.18.E10 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

and two similar equations obtained by permuting $x, y, z$ in (19.18.10).

More concisely, if $v=R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ , 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: $\,\partial\NVar{x}$ : partial differential of $x$ , $n$ : nonnegative integer and $v$ Referenced by: §19.18(ii), §19.18(ii) Permalink: http://dlmf.nist.gov/19.18.E11 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

and also a system of $n(n-1)/2$ Euler–Poisson differential equations (of which only $n-1$ are independent):

19.18.12		$(z_{j}\partial_{j}+b_{j})\partial_{l}v=(z_{l}\partial_{l}+b_{l})\partial_{j}v,$
ⓘ Symbols: $\,\partial\NVar{x}$ : partial differential of $x$ , $l$ : nonnegative integer and $v$ Referenced by: §19.18(ii), §19.18(ii) Permalink: http://dlmf.nist.gov/19.18.E12 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

or equivalently,

19.18.13		$((z_{j}-z_{l})\partial_{j}\partial_{l}+b_{j}\partial_{l}-b_{l}\partial_{j})v=0.$
ⓘ Symbols: $\,\partial\NVar{x}$ : partial differential of $x$ , $l$ : nonnegative integer and $v$ Permalink: http://dlmf.nist.gov/19.18.E13 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

Here $j,l=1,2,\dots,n$ and $j\neq l$ . For group-theoretical aspects of this system see Carlson (1963, §VI). If $n=2$ , 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 $R_{F}$ and $R_{G}$ in the form $R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)$ (see (19.16.20) and (19.16.23)).

The function $w=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+y,x-y\right)$ 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: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$ Permalink: http://dlmf.nist.gov/19.18.E14 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

Also $W=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};t+r,t-r\right)$ , 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: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $W$ : wave function Permalink: http://dlmf.nist.gov/19.18.E15 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

Similarly, the function $u=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+iy,x-iy\right)$ 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: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$ Permalink: http://dlmf.nist.gov/19.18.E16 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

and $U=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z+i\rho,z-i\rho\right)$ , 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: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $U$ Referenced by: §19.18(ii) Permalink: http://dlmf.nist.gov/19.18.E17 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

19.17 Graphics 19.19 Taylor and Related Series

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§19.18 Derivatives and Differential Equations

Contents

§19.18(i) Derivatives

§19.18(ii) Differential Equations