lattice parameter

SearchAdvancedHelp

(0.001 seconds)

11—19 of 19 matching pages

11: 20.3 Graphics

… ►

§20.3(iii) $\theta$-Functions: Real Variable and Complex Lattice Parameter

…

12: 31.2 Differential Equations

… ►

31.2.10 $$w(\xi )={\left(\mathrm{\wp }\left(\xi \right)-e_3\right)}^{(1-2\gamma )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_2\right)}^{(1-2\delta )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_1\right)}^{(1-2ϵ)/4}W(\xi ),$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝕃$: lattice and $W(\xi )$: solution

Permalink:

http://dlmf.nist.gov/31.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

…

13: 29.2 Differential Equations

… ►

29.2.9 $$\frac{d^{2}w}{{d\eta }^2}+(g-\nu (\nu +1)\mathrm{\wp }\left(\eta \right))w=0,$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\nu$: real parameter, $g$ and $\eta$

Permalink:

http://dlmf.nist.gov/29.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

…

14: 36.7 Zeros

… ►

36.7.4 $$z_n=\pm 3{(\frac{1}{4}\pi (2n-\frac{1}{2}))}^{1/3}=3.48734{(n-\frac{1}{4})}^{1/3},$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: real parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/36.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.7(iii), §36.7 and Ch.36

►Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by … ►

36.7.6 $$\exp \left(-2\pi \mathrm{i}\left(\frac{z-z_n}{\Delta z}+\frac{2x}{\Delta x}\right)\right)\left(2\exp \left(\frac{-6\pi \mathrm{i}x}{\Delta x}\right)\cos \left(\frac{2\sqrt{3}\pi y}{\Delta x}\right)+1\right)=\sqrt{3}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.7(iii), §36.7 and Ch.36

… ►Away from the $z$-axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. …

15: 23.20 Mathematical Applications

… ►

Rectangular Lattice

… ►

Rhombic Lattice

… ►For each pair of edges there is a unique point $z_0$ such that $\mathrm{\wp }\left(z_0\right)=0$. … ►Points $P=(x,y)$ on the curve can be parametrized by $x=\mathrm{\wp }(z;g_2,g_3)$, $2y={\mathrm{\wp }}^{\prime }(z;g_2,g_3)$, where $g_2=-4a$ and $g_3=-4b$: in this case we write $P=P(z)$. … ►These cases correspond to rhombic and rectangular lattices, respectively. …

16: 31.17 Physical Applications

… ►for the common eigenfunction $\mathrm{\Psi }(𝐱)=\mathrm{\Psi }(x_s,x_t,x_u)$, where $a$ is the coupling parameter of interacting spins. … ►

31.17.2 $$\frac{x_s^2}{z_k}+\frac{x_t^2}{z_k-1}+\frac{x_u^2}{z_k-a}=0,$$ $k=1,2$,

ⓘ

Symbols:

$z$: complex variable, $a$: complex parameter and $x_s$, $x_t$, $x_u$: coordinates

Permalink:

http://dlmf.nist.gov/31.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

… ►

31.17.4 $$\mathrm{\Psi }(𝐱)={(z_1{z}_2)}^{-s-\frac{1}{4}}{((z_1-1)(z_2-1))}^{-t-\frac{1}{4}}{((z_1-a)(z_2-a))}^{-u-\frac{1}{4}}w(z_1)w(z_2),$$

ⓘ

Symbols:

$z$: complex variable, $a$: complex parameter and $\mathrm{\Psi }(𝐱)$: common eigenfunction

Permalink:

http://dlmf.nist.gov/31.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

►where $w(z)$ satisfies Heun’s equation (31.2.1) with $a$ as in (31.17.1) and the other parameters given by … ►Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). …

17: 26.9 Integer Partitions: Restricted Number and Part Size

… ►It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. … ►It is also equal to the number of lattice paths from $(0,0)$ to $(m,k)$ that have exactly $n$ vertices $(h,j)$, $1\le h\le m$, $1\le j\le k$, above and to the left of the lattice path. … ►

Figure 26.9.2: The partition $5+5+3+2$ represented as a lattice path.

… ►

26.9.4 $${\left[\genfrac{}{}{0.0pt}{}{m}{n}\right]}_q=\prod _{j=1}^n\frac{1-q^{m-n+j}}{1-q^j},$$ $n\ge 0$,

ⓘ

Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

… ►It is also assumed everywhere that . …

18: 18.19 Hahn Class: Definitions

… ►

Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

►

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 $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

… ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

► ►►

	$p_n(x)$	$X$	$w_x$	$h_n$
…

►

…

19: 18.27 $q$-Hahn Class

… ►The $q$-hypergeometric OP’s comprise the $q$-Hahn class (or $q$-linear lattice class) OP’s and the Askey–Wilson class (or $q$-quadratic lattice class) OP’s (§18.28). … ►These families depend on further parameters, in addition to $q$. The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters. …