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1: Bibliography Y

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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External Links:

MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

§35.10.

Encodings:

BibTeX

…

2: 19.21 Connection Formulas

… ►

19.21.1 $$R_F(0,z+1,z)R_D(0,z+1,1)+R_D(0,z+1,z)R_F(0,z+1,1)=3\pi /(2z),$$ $z\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval and $\setminus$: set subtraction

Referenced by:

§19.21(i), §19.21(i), §19.7(i)

Permalink:

http://dlmf.nist.gov/19.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

… ►

19.21.2 $$3R_F(0,y,z)=z{R}_D(0,y,z)+y{R}_D(0,z,y).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.21(i)

Permalink:

http://dlmf.nist.gov/19.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

►

19.21.3 $$6R_G(0,y,z)=yz(R_D(0,y,z)+R_D(0,z,y))=3z{R}_F(0,y,z)+z(y-z)R_D(0,y,z).$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.21(i), §19.22(i), §19.34

Permalink:

http://dlmf.nist.gov/19.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

… ►

19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

Encodings:

pMML, png, TeX

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 §19.21(ii), §19.21 and Ch.19

►

19.21.9 $$x{R}_D(y,z,x)+y{R}_D(z,x,y)+z{R}_D(x,y,z)=3R_F(x,y,z).$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.21(i), §19.21(ii)

Permalink:

http://dlmf.nist.gov/19.21.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

…

3: 16.13 Appell Functions

§16.13 Appell Functions

►The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${}_{2}F_1$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …

4: 19.27 Asymptotic Approximations and Expansions

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19.27.7 $$R_D(x,y,z)=\frac{3}{2z^{3/2}}\left(\ln \left(\frac{8z}{a+g}\right)-2\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.8 $$R_D(x,y,z)=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}{R}_G(x,y,0)\left(1+O\left(\frac{z}{g}\right)\right),$$ $z/g\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $g$

Referenced by:

§19.20(iv), §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.27.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.9 $$R_D(x,y,z)=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+O\left(\frac{b}{x}\ln \frac{x}{b}\right)\right),$$ $b/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function and $b$

Permalink:

http://dlmf.nist.gov/19.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.10 $$R_D(x,y,z)=R_D(0,y,z)-\frac{3\sqrt{x}}{hz}\left(1+O\left(\sqrt{\frac{x}{h}}\right)\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $h$

Permalink:

http://dlmf.nist.gov/19.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

…

5: 16.1 Special Notation

… ►The main functions treated in this chapter are the generalized hypergeometric function ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$, the Appell (two-variable hypergeometric) functions $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, and the Meijer $G$-function $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$. …

6: 19.20 Special Cases

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19.20.19 $$R_D(x,y,z)\sim 3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$ $z/\sqrt{xy}\to 0$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\sim$: asymptotic equality

Referenced by:

§19.20(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.20.E19

Encodings:

pMML, png, TeX

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 §19.20(iv), §19.20 and Ch.19

►

19.20.20 $$R_D(x,y,y)=\frac{3}{2(y-x)}\left(R_C(x,y)-\frac{\sqrt{x}}{y}\right),$$ $x\ne y$, $y\ne 0$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.20.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

►

19.20.21 $$R_D(x,x,z)=\frac{3}{z-x}\left(R_C(z,x)-\frac{1}{\sqrt{z}}\right),$$ $x\ne z$, $xz\ne 0$.

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.20.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

… ►

19.20.22 $${\int }_0^1\frac{t^{2}dt}{\sqrt{1-t^4}}=\frac{1}{3}{R}_D(0,2,1)=\frac{{\left(\mathrm{\Gamma }\left(\frac{3}{4}\right)\right)}^2}{{(2\pi )}^{1/2}}=\mathrm{0.59907\hspace{0.33em}01173\hspace{0.33em}67796\hspace{0.33em}10371}\mathrm{\dots }.$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential and $\int$: integral

Notes:

For more digits see OEIS Sequence A085566; see also Sloane (2003).

Referenced by:

§19.20(iv), §19.21(i), Figure 19.3.6, Figure 19.3.6, Figure 19.3.6

Permalink:

http://dlmf.nist.gov/19.20.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

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19.20.23 $$R_D(x,y,a)=R_{-\frac{3}{4}}(\frac{5}{4},\frac{1}{2};a^2,xy),$$ $a=\frac{1}{2}x+\frac{1}{2}y$.

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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 and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.20.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

…

7: 19.28 Integrals of Elliptic Integrals

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19.28.3 $${\int }_0^1{t}^{\sigma -1}(1-t)R_D(0,t,1)dt=\frac{3}{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2.$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{B}(a,b)$: beta function, $dx$: differential, $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

… ►

19.28.5 $${\int }_z^{\mathrm{\infty }}{R}_D(x,y,t)dt=6R_F(x,y,z),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.6 $${\int }_0^1{R}_D(x,y,v^{2}z+(1-v^2)p)dv=R_J(x,y,z,p).$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

…

8: Viewing DLMF Interactive 3D Graphics

… ►In the DLMF, we provide facilities for the interactive display of special functions of two independent variables. …

9: 19.16 Definitions

… ►A fourth integral that is symmetric in only two variables is defined by ►

19.16.5 $$R_D(x,y,z)=R_J(x,y,z,z)=\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)(t+z)},$$

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Defines:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Symbols:

$R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential, $\int$: integral and $s(t)$: scaling factor

Referenced by:

§19.18(i), §19.19, §19.20(iii), §19.20(iv), §19.24(ii), §19.6(iv)

Permalink:

http://dlmf.nist.gov/19.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

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19.16.15 $$R_D(x,y,z)=R_{-\frac{3}{2}}(\frac{1}{2},\frac{1}{2},\frac{3}{2};x,y,z),$$

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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 and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.16.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

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19.16.21 $$R_D(0,y,z)=\frac{3}{4}\pi R_{-\frac{3}{2}}(\frac{1}{2},\frac{3}{2};y,z),$$

ⓘ

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, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\pi$: the ratio of the circumference of a circle to its diameter

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.16.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

…

10: 19.25 Relations to Other Functions

… ►

19.25.6 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{1}{3}k{R}_D(c-1,c,c-k^2).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

… ►

19.25.9 $$E(\varphi ,k)=R_F(c-1,c-k^2,c)-\frac{1}{3}{k}^2{R}_D(c-1,c-k^2,c),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.25(i), §19.25(iii), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.25.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

… ►

19.25.12 $$\frac{\partial E(\varphi ,k)}{\partial k}=-\frac{1}{3}k{R}_D(c-1,c-k^2,c).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

►

19.25.13 $$D(\varphi ,k)=\frac{1}{3}{R}_D(c-1,c-k^2,c).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.25.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.21 $$\mathrm{el2}(x,k_c,a,b)=a\mathrm{el1}(x,k_c)+\frac{1}{3}(b-a)R_D(r,r+k_c^2,r+1),$$

ⓘ

Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el2}(x,k_c,a,b)$: Bulirsch’s incomplete elliptic integral of the second kind, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

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