# §19.18 Derivatives and Differential Equations

## §19.18(i) Derivatives

 19.18.1 $\frac{\partial R_{F}\left(x,y,z\right)}{\partial z}=-\tfrac{1}{6}R_{D}\left(x,% y,z\right),$
 19.18.2 $\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x+c\right)=\tfrac{1}{2}R_{F}% \left(x+a,x+b,x+c\right).$

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 $\displaystyle w_{j}$ $\displaystyle=b_{j}\biggm{/}\sum_{j=1}^{n}b_{j},$ $\displaystyle a^{\prime}$ $\displaystyle=-a+\sum_{j=1}^{n}b_{j}.$ ⓘ 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 $\partial_{j}R_{-a}\left(\mathbf{b};\mathbf{z}\right)=-aw_{j}R_{-a-1}\left(% \mathbf{b}+\mathbf{e}_{j};\mathbf{z}\right),$
 19.18.5 $(z_{j}\partial_{j}+b_{j})R_{-a}\left(\mathbf{b};\mathbf{z}\right)=w_{j}a^{% \prime}R_{-a}\left(\mathbf{b}+\mathbf{e}_{j};\mathbf{z}\right).$

## §19.18(ii) Differential Equations

 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 and $\partial\NVar{x}$: partial differential 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 $\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)R_{G}\left(x,y,z\right)=\tfrac{1}{2}R_{F}\left(x,y,z\right).$
 19.18.8 $\sum_{j=1}^{n}\partial_{j}R_{-a}\left(\mathbf{b};\mathbf{z}\right)=-aR_{-a-1}% \left(\mathbf{b};\mathbf{z}\right).$
 19.18.9 $\left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{% \partial}{\partial z}\right)R_{F}\left(x,y,z\right)=-\tfrac{1}{2}R_{F}\left(x,% y,z\right),$
 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 and $\partial\NVar{x}$: partial differential 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, $n$: 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, $l$: 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, $l$: 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 and $\partial\NVar{x}$: partial differential 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, $\partial\NVar{x}$: partial differential 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 and $\partial\NVar{x}$: partial differential 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, $\partial\NVar{x}$: partial differential 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