in two or more variables

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11: 18.40 Methods of Computation

… ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …

12: 9.16 Physical Applications

… ►The function

\operatorname{Ai}\left(x\right)

first appears as an integral in two articles by G. … ►In the study of the stability of a two-dimensional viscous fluid, the flow is governed by the Orr–Sommerfeld equation (a fourth-order differential equation). …

13: 19.16 Definitions

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

19.16.5

R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\,\mathrm{d}t}{s(t)(t+z)},

ⓘ

Defines:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Symbols:: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : 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

… ►

19.16.15

R_{D}\left(x,y,z\right)=R_{-\frac{3}{2}}\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac% {3}{2};x,y,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 and $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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

… ►

19.16.21

R_{D}\left(0,y,z\right)=\tfrac{3}{4}\pi R_{-\frac{3}{2}}\left(\tfrac{1}{2},% \tfrac{3}{2};y,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, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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

…

14: 19.23 Integral Representations

… ►

19.23.3

R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)% ^{-3/2}{\sin}^{2}\theta\,\mathrm{d}\theta.

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

… ►In (19.23.8)–(19.23.10) one or more of the variables may be 0 if the integral converges. …

15: 29.7 Asymptotic Expansions

… ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as

\nu\to\infty

, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …

16: 35.2 Laplace Transform

… ►where the integration variable

\mathbf{X}

ranges over the space

{\boldsymbol{\Omega}}

. ►Suppose there exists a constant

\mathbf{X}_{0}\in{\boldsymbol{\Omega}}

such that

|f(\mathbf{X})|<\operatorname{etr}\left(-\mathbf{X}_{0}\mathbf{X}\right)

for all

\mathbf{X}\in{\boldsymbol{\Omega}}

. …

17: 19.29 Reduction of General Elliptic Integrals

… ►

19.29.7

\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{\delta}+b_{\delta}t}\frac{\,% \mathrm{d}t}{s(t)}=\tfrac{2}{3}d_{\alpha\beta}d_{\alpha\gamma}R_{D}\left(U_{% \alpha\beta}^{2},U_{\alpha\gamma}^{2},U_{\alpha\delta}^{2}\right)+\frac{2X_{% \alpha}Y_{\alpha}}{X_{\delta}Y_{\delta}U_{\alpha\delta}},

U_{\alpha\delta}\neq 0

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $X_{\alpha}$ , $U_{n}$ , $Y_{\alpha}$ , $d_{\alpha\beta}$ , $s(t)$ : product, $\alpha$ : parameter, $\beta$ : parameter, $\gamma$ : parameter and $\delta$ : parameter
Referenced by:: §19.29(i), §19.29(i), §19.29(ii), §19.29(ii), §19.29(iii), §19.30(ii), §19.34
Permalink:: http://dlmf.nist.gov/19.29.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(i), §19.29 and Ch.19

… ►

19.29.10

\int_{u}^{b}\sqrt{\frac{a-t}{(b-t)(t-c)^{3}}}\,\mathrm{d}t=-\tfrac{2}{3}{(a-b)% }{(b-u)}^{3/2}R_{D}+\frac{2}{b-c}\sqrt{\frac{(a-u)(b-u)}{u-c}},

a>b>u>c

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.29(i)
Permalink:: http://dlmf.nist.gov/19.29.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(i), §19.29 and Ch.19

… ►

19.29.20

\int_{y}^{x}\frac{t^{2}\,\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}a_{% 1}a_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)+(xy/U),

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $U$ and $Q(t^{2})$
Referenced by:: §19.20(iv), §19.25(v), §19.29(iii), §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(iii), §19.29 and Ch.19

… ►

19.29.21

\int_{y}^{x}\frac{\,\mathrm{d}t}{t^{2}\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}b_{% 1}b_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)+(xyU)^{-1},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $U$ and $Q(t^{2})$
Referenced by:: §19.25(v), §19.29(iii), §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(iii), §19.29 and Ch.19

… ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as