in two or more variables

(0.001 seconds)

1—10 of 302 matching pages

1: 18.37 Classical OP’s in Two or More Variables

§18.37 Classical OP’s in Two or More Variables

…

2: 37.2 General Orthogonal Polynomials of Two Variables

… ►In this section polynomials will be polynomials of two variables with real coefficients. …A weight function

W

is a nonnegative function on an open set

\Omega\subset{\mathbb{R}}^{2}

such that the integral

\int_{\Omega}P(x,y)W(x,y)\,\mathrm{d}x\,\mathrm{d}y

is well-defined and absolutely convergent for all polynomials

P

, and such that

\int_{\Omega}W(x,y)\,\mathrm{d}x\,\mathrm{d}y>0

. …The space

\mathcal{V}_{n}

of orthogonal polynomials of degree $n$ consists of all

P\in\Pi_{n}

such that

\left\langle P,Q\right\rangle_{W}=0

for all

Q\in\Pi_{n-1}

(

n>0

, otherwise

\mathcal{V}_{0}=\Pi_{0}

). … ►See Dunkl and Xu (2014, Theorem 3.3.8 and §3.2,

d=2

) for this theorem and for the definitions involved in its formulation. … ►

37.2.28

L=A(x,y){D}_{xx}+B(x,y){D}_{xy}+C(x,y){D}_{yy}+D(x,y){D}_{x}+E(x,y){D}_{y}

ⓘ

Defines:: $L$ : second order partial differential operator in two variables (locally), $A$ : function in two variables (locally), $B$ : function in two variables (locally), $C$ : function in two variables (locally), $D$ : function in two variables (locally) and $E$ : function in two variables (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/37.2.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §37.2(viii), §37.2 and Ch.37

…

3: Mark J. Ablowitz

… ►Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. …

4: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

… ►

37.3.1

W_{\alpha,\beta,\gamma}(x,y)=x^{\alpha}y^{\beta}(1-x-y)^{\gamma}

ⓘ

Defines:: $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ (locally)
Symbols:: $\triangle$ : triangular region, $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Referenced by:: §37.5(iii)
Permalink:: http://dlmf.nist.gov/37.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(i), §37.3 and Ch.37

… ►

37.3.14

\left[x(1-x){D}_{xx}-2xy{D}_{xy}+y(1-y){D}_{yy}+(\alpha+1-(\alpha+\beta+\gamma% +3)x){D}_{x}+(\beta+1-(\alpha+\beta+\gamma+3)y){D}_{y}\right]{}u(x,y)=-n(n+% \alpha+\beta+\gamma+2)u(x,y),

u\in\mathcal{V}_{n}^{\alpha,\beta,\gamma}

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\in$ : element of, $n$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ), $\gamma$ : real parameter ( $>-1$ ), $\mathcal{V}_{n}^{\alpha,\beta,\gamma}$ : vector space of polynomials of degree $n$ on the triangular region and $u$ : real variable
Proved:: Dunkl and Xu (2014, (5.3.4) and Theorem 8.2.1); for $d=2$ , $\kappa_{1}=\alpha+\frac{1}{2}$ , $\kappa_{2}=\beta+\frac{1}{2}$ , $\mu=\gamma+\frac{1}{2}$ (proved)
Referenced by:: item 1, (37.3.15), §37.3(i), §37.3(iv), §37.7(iii)
Permalink:: http://dlmf.nist.gov/37.3.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(iv), §37.3 and Ch.37

… ►

37.3.15

W_{\alpha,\beta,\gamma}(x,y)^{-1}\Big({D}_{x}[W_{\alpha+1,\beta,\gamma+1}(x,y)% {D}_{x}u(x,y)]+{D}_{y}[W_{\alpha,\beta+1,\gamma+1}(x,y){D}_{y}u(x,y)]+{D}_{z}[% W_{\alpha+1,\beta+1,\gamma}(x,y){D}_{z}u(x,y)]\Big)=-n(n+\alpha+\beta+\gamma+2% )u(x,y),

u\in\mathcal{V}_{n}

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\in$ : element of, $n$ : nonnegative integer, $z$ : complex number, $x$ : real variable, $y$ : real variable, $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ , $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ), $\gamma$ : real parameter ( $>-1$ ), $\mathcal{V}_{n}$ : vector space of two-variable polynomials of degree $n$ on triangular region and $u$ : real variable
Proof sketch:: By direct computation the left-hand side of the present equality equals the left-hand side of (37.3.14).
Permalink:: http://dlmf.nist.gov/37.3.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(iv), §37.3 and Ch.37

… ►

37.3.24

{D}_{x}\left(W_{\alpha+1,\beta,\gamma+1}(x,y)U^{\alpha+1,\beta,\gamma+1}_{k-1,% n-1}\left(x,y\right)\right)=W_{\alpha,\beta,\gamma}(x,y)U^{\alpha,\beta,\gamma% }_{k,n}\left(x,y\right),

ⓘ

Symbols:: $U^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : co-monic basis element of two-variable Jacobi polynomial on the triangle, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer, $k$ : nonnegative integer, $x$ : real variable, $y$ : real variable, $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ , $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Proof sketch:: Use the first equality in (37.3.12).
Permalink:: http://dlmf.nist.gov/37.3.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(v), §37.3 and Ch.37

… ►

37.3.26

{D}_{y}\left(W_{\alpha,\beta+1,\gamma+1}(x,y)U^{\alpha,\beta+1,\gamma+1}_{k,n-% 1}\left(x,y\right)\right)=W_{\alpha,\beta,\gamma}(x,y)U^{\alpha,\beta,\gamma}_% {k,n}\left(x,y\right).

ⓘ

Symbols:: $U^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : co-monic basis element of two-variable Jacobi polynomial on the triangle, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer, $k$ : nonnegative integer, $x$ : real variable, $y$ : real variable, $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ , $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Proof sketch:: Use the first equality in (37.3.12).
Permalink:: http://dlmf.nist.gov/37.3.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(v), §37.3 and Ch.37

…

5: 1.5 Calculus of Two or More Variables

§1.5 Calculus of Two or More Variables

… ►

Z. M. Yan (1992) 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.

ⓘ

External Links:: MathReview (B. M. Agrawal), ZentralBlatt
Referenced by:: §35.10.
Encodings:: BibTeX

…

7: 8.13 Zeros

… ►

(a)

two zeros in each of the intervals $-2n<a<2-2n$ when $x<0$ ;

►

(b)

two zeros in each of the intervals $-2n<a<1-2n$ when $0<x\leq x_{n}^{*}$ ;

…

8: 28.33 Physical Applications

… ►Mathieu functions occur in practical applications in two main categories: ►

•

Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

…

9: Sidebar 21.SB1: Periodic Surface Waves

… ►Two-dimensional periodic waves in a shallow water wave tank. Taken from Joe Hammack, Daryl McCallister, Norman Scheffner and Harvey Segur, “Two-dimensional periodic waves in shallow water. …The caption reads “Mosaic of two overhead photographs, showing surface patterns of waves in shallow water”. …

10: 19.27 Asymptotic Approximations and Expansions

… ►

19.27.7

R_{D}\left(x,y,z\right)=\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:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\ln\NVar{z}$ : 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}\left(x,y,z\right)=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}R_{G}\left(x,y,0% \right)\left(1+O\left(\frac{z}{g}\right)\right),

z/g\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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}\left(x,y,z\right)=\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:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\ln\NVar{z}$ : 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

… ►The approximations in §§19.27(i)–19.27(v) are furnished with upper and lower bounds by Carlson and Gustafson (1994), sometimes with two or three approximations of differing accuracies. … ►A similar (but more general) situation prevails for

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

when some of the variables

z_{1},\dots,z_{n}

are smaller in magnitude than the rest; see Carlson (1985, (4.16)–(4.19) and (2.26)–(2.29)). …