two or more variables

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11—20 of 286 matching pages

11: Guide to Searching the DLMF

Table 1: Query Examples

Query	Matching records contain
…
`sin x and (J_nu(z) or I_nu(z))`	both $\sin x$ and at least one of the two functions $J_{\nu}(z)$ or $I_{\nu}(z)$ .
…
`trigonometric^2 + trig$^2`	any sum of the squares of two trigonometric functions such as ${\sin}^{2}z+{\cos}^{2}z$ .
…

â–º … â–ºDLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. …

12: 29.19 Physical Applications

… â–º

§29.19(ii) Lamé Polynomials

… â–ºMacfadyen and Winternitz (1971) finds expansions for the two-body relativistic scattering amplitudes. …

13: Bibliography Q

… â–º

S.-L. Qiu and J.-M. Shen (1997) On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).

â“˜

Notes:: In Chinese
Referenced by:: §19.9(i).
Encodings:: BibTeX

… â–º

W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

â“˜

External Links:: ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt
Referenced by:: §2.4(vi).
Encodings:: BibTeX

…

14: Bibliography I

… â–º

M. E. H. Ismail, J. Letessier, G. Valent, and J. Wimp (1990) Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.

â“˜

External Links:: ISSN 0008-414X, MathReview (Mizan Rahman), ZentralBlatt
Referenced by:: §18.26(iv).
Encodings:: BibTeX

… â–º

M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

â“˜

External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

… â–º

M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

â“˜

Notes:: With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.
External Links:: ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt
Referenced by:: §18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

…

15: 19.27 Asymptotic Approximations and Expansions

… â–º

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

â“˜

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

â–º

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

â“˜

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

â–º

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

â“˜

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

… â–ºThe 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. … â–ºA 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)). …

16: 31.4 Solutions Analytic at Two Singularities: Heun Functions

§31.4 Solutions Analytic at Two Singularities: Heun Functions

…

17: 31.8 Solutions via Quadratures

… â–ºare two independent solutions of (31.2.1). Here

\Psi_{g,N}(\lambda,z)

is a polynomial of degree

g

\lambda

and of degree

N=m_{0}+m_{1}+m_{2}+m_{3}

z

, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. …The variables

\lambda

and

\nu

are two coordinates of the associated hyperelliptic (spectral) curve

\Gamma:\nu^{2}=\prod_{j=1}^{2g+1}(\lambda-\lambda_{j})

. …

18: Frank Garvan

… â–ºHe has written two books on MAPLE. …

19: 8.13 Zeros

… â–º

(a)

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

â–º

(b)

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

…

20: Mathematical Introduction

… â–ºTwo other ways in which this Handbook differs from AMS 55, and other handbooks, are as follows. … â–ºThese include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … â–ºSpecial functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … â–ºWith two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … â–ºSpecial functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …