§19.18 Derivatives and Differential Equations

19.18.1		$$\frac{\partial R_F(x,y,z)}{\partial z}=-\frac{1}{6}{R}_D(x,y,z),$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: 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		$$\frac{d}{dx}{R}_G(x+a,x+b,x+c)=\frac{1}{2}{R}_F(x+a,x+b,x+c).$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\frac{df}{dx}$: 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 ${\partial }_j=\partial /\partial z_j$, and ${\mathbf{e}}_j$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. Also define

19.18.3		$w_j$	$=b_j/{\displaystyle \sum _{j=1}^n}{b}_j,$
		$a^{\prime }$	$=-a+{\displaystyle \sum _{j=1}^n}{b}_j.$
ⓘ 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		$${\partial }_j{R}_{-a}(\mathbf{b};\mathbf{z})=-a{w}_j{R}_{-a-1}(\mathbf{b}+{\mathbf{e}}_j;\mathbf{z}),$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\partial x$: 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		$$(z_j{\partial }_j+b_j)R_{-a}(\mathbf{b};\mathbf{z})=w_j{a}^{\prime }{R}_{-a}(\mathbf{b}+{\mathbf{e}}_j;\mathbf{z}).$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\partial x$: 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

19.18.6		$$\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)R_F(x,y,z)=\frac{-1}{2\sqrt{xyz}},$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$ Referenced by: §19.18(ii), §19.32 Permalink: http://dlmf.nist.gov/19.18.E6 Encodings: TeX, pMML, png See also: Annotations for §19.18(ii), §19.18 and Ch.19

19.18.7		$$\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)R_G(x,y,z)=\frac{1}{2}{R}_F(x,y,z).$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: 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		$$\sum _{j=1}^n{\partial }_j{R}_{-a}(\mathbf{b};\mathbf{z})=-a{R}_{-a-1}(\mathbf{b};\mathbf{z}).$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\partial x$: 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		$$\left(x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+z\frac{\partial }{\partial z}\right)R_F(x,y,z)=-\frac{1}{2}{R}_F(x,y,z),$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: 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		$$\left((x-y)\frac{{\partial }^2}{\partial x\partial y}+\frac{1}{2}\left(\frac{\partial }{\partial y}-\frac{\partial }{\partial x}\right)\right)R_F(x,y,z)=0,$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: 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 $v=R_{-a}(\mathbf{b};\mathbf{z})$, 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 $x$, $n$: 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: $\partial x$: 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: $\partial x$: 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 $j,l=1,2,\mathrm{\dots },n$ and $j\ne l$. 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 $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)).

The function $w=R_{-a}(\frac{1}{2},\frac{1}{2};x+y,x-y)$ 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 $f$ with respect to $x$ and $\partial x$: 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 $W=R_{-a}(\frac{1}{2},\frac{1}{2};t+r,t-r)$, 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 $f$ with respect to $x$, $\partial x$: 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 $u=R_{-a}(\frac{1}{2},\frac{1}{2};x+\mathrm{i}y,x-\mathrm{i}y)$ 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 $f$ with respect to $x$ and $\partial x$: 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 $U=R_{-a}(\frac{1}{2},\frac{1}{2};z+\mathrm{i}\rho ,z-\mathrm{i}\rho )$, 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 $f$ with respect to $x$, $\partial x$: 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