in two or more variables

(0.008 seconds)

21—30 of 282 matching pages

21: 31.17 Physical Applications

… ►The problem of adding three quantum spins

\mathbf{s}

\mathbf{t}

, and

\mathbf{u}

can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … ►For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986). … ►More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). …

22: 19.22 Quadratic Transformations

… ►

19.22.3

2y^{2}R_{D}\left(0,x^{2},y^{2}\right)=\tfrac{1}{4}(y^{2}-x^{2})R_{D}\left(0,xy% ,a^{2}\right)+3R_{F}\left(0,xy,a^{2}\right).

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(i), §19.22(i), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(i), §19.22 and Ch.19

… ►The AGM,

M\left(a_{0},g_{0}\right)

, of two positive numbers

a_{0}

and

g_{0}

is defined in §19.8(i). … ►

19.22.10

R_{D}\left(0,g_{0}^{2},a_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_{0}\right)% a_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.22(ii)
Permalink:: http://dlmf.nist.gov/19.22.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

… ►

19.22.19

(z_{\pm}^{2}-z_{\mp}^{2})R_{D}\left(x^{2},y^{2},z^{2}\right)={2(z_{\pm}^{2}-a^% {2})}R_{D}\left(a^{2},z_{\mp}^{2},z_{\pm}^{2}\right)-3R_{F}\left(x^{2},y^{2},z% ^{2}\right)+(3/z),

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(iii), §19.22(iii), §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.22.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(iii), §19.22 and Ch.19

…

23: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►Such a solution is given in terms of a Riemann theta function with two phases. …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

24: 13.5 Continued Fractions

… ►If

a,b\in\mathbb{C}

such that

a\neq-1,-2,-3,\dots

, and

a-b\neq 0,1,2,\dots

, then ►

13.5.1

\frac{M\left(a,b,z\right)}{M\left(a+1,b+1,z\right)}=1+\cfrac{u_{1}z}{1+\cfrac{% u_{2}z}{1+\cdots}},

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $z$ : complex variable and $u_{n}$ : continued fraction coefficients
Referenced by:: §13.31(ii)
Permalink:: http://dlmf.nist.gov/13.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.5 and Ch.13

… ►This continued fraction converges to the meromorphic function of

z

on the left-hand side everywhere in

\mathbb{C}

. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). ►If

a,b\in\mathbb{C}

such that

a\neq 0,-1,-2,\dots

, and

b-a\neq 2,3,4,\dots

, then …

25: 20.12 Mathematical Applications

… ►The space of complex tori

\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})

(that is, the set of complex numbers

z

in which two of these numbers

z_{1}

and

z_{2}

are regarded as equivalent if there exist integers

m, n

such that

z_{1}-z_{2}=m+\tau n

) is mapped into the projective space

P^{3}

via the identification

z\to(\theta_{1}\left(2z\middle|\tau\right),\theta_{2}\left(2z\middle|\tau% \right),\theta_{3}\left(2z\middle|\tau\right),\theta_{4}\left(2z\middle|\tau% \right))

. …

26: Bibliography R

… ►

J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

ⓘ

External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

27: 1.10 Functions of a Complex Variable

… ►Branches can be constructed in two ways: …

28: 21.8 Abelian Functions

… ►In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

29: 18.19 Hahn Class: Definitions

… ►These eight further families can be grouped in two classes of OP’s: … ►

Wilson class (or quadratic lattice class). These are OP’s $p_{n}(x)=p_{n}(\lambda(y))$ ( $p_{n}(x)$ of degree $n$ in $x$ , $\lambda(y)$ quadratic in $y$ ) where the role of the differentiation operator is played by $\tfrac{\Delta_{y}}{\Delta_{y}(\lambda(y))}$ or $\tfrac{\nabla_{y}}{\nabla_{y}(\lambda(y))}$ or $\tfrac{\delta_{y}}{\delta_{y}(\lambda(y))}$ . The Wilson class consists of two discrete and two continuous families.

…

30: Errata

… ►

Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

… ►

Notation

The symbol $\sim$ is used for two purposes in the DLMF, in some cases for asymptotic equality and in other cases for asymptotic expansion, but links to the appropriate definitions were not provided. In this release changes have been made to provide these links.

►

Subsection 2.1(iii)

A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

… ►

Equations (28.28.21) and (28.28.22)

28.28.21

\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\cos\left((2\ell% +1)\phi\right)\operatorname{ce}_{2m+1}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}A^{2m+1}_{2\ell+1}(h^{2}){\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)

28.28.22

\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\sin\left((2\ell% +1)\phi\right)\operatorname{se}_{2m+1}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}B^{2m+1}_{2\ell+1}(h^{2}){\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h% \right),

Originally the prefactor $\frac{4}{\pi}$ and upper limit of integration $\pi/2$ in these two equations were given incorrectly as $\frac{2}{\pi}$ and $\pi$ .

Reported 2015-05-20 by Ruslan Kabasayev

… ►

Paragraph Case III:

V(x)=-\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}

(in §22.19(ii))

Two corrections have been made in this paragraph. First, the correct range of the initial displacement $a$ is $\sqrt{1/\beta}\leq|a|<\sqrt{2/\beta}$ . Previously it was $\sqrt{1/\beta}\leq|a|\leq\sqrt{2/\beta}$ . Second, the correct period of the oscillations is $2\!K\left(k\right)/\sqrt{\eta}$ . Previously it was given incorrectly as $4\!K\left(k\right)/\sqrt{\eta}$ .

Reported 2014-05-02 by Svante Janson.

…