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21: 19.35 Other Applications

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§19.35(ii) Physical

►Elliptic integrals appear in lattice models of critical phenomena (Guttmann and Prellberg (1993)); theories of layered materials (Parkinson (1969)); fluid dynamics (Kida (1981)); string theory (Arutyunov and Staudacher (2004)); astrophysics (Dexter and Agol (2009)). …

22: 23.22 Methods of Computation

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§23.22(ii) Lattice Calculations

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Starting from Lattice

►Suppose that the lattice $𝕃$ is given. …The corresponding values of $e_1$, $e_2$, $e_3$ are calculated from (23.6.2)–(23.6.4), then $g_2$ and $g_3$ are obtained from (23.3.6) and (23.3.7). … ►Suppose that the invariants $g_2=c$, $g_3=d$, are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). …

23: 19.25 Relations to Other Functions

… ►Let $𝕃$ be a lattice for the Weierstrass elliptic function $\mathrm{\wp }\left(z\right)$. … ►

19.25.36 $$\mathrm{\wp }\left(z\right)-e_j\in ℂ\setminus (-\mathrm{\infty },0],$$ $j=1,2,3.$

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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, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $j$: $(=1,2,3)$ and $z$: complex number

Referenced by:

(19.25.39)

Permalink:

http://dlmf.nist.gov/19.25.E36

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Annotations for §19.25(vi), §19.25 and Ch.19

►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which , for some $j$. … ►for some $2{\omega }_j\in 𝕃$ and $\mathrm{\wp }\left({\omega }_j\right)=e_j$. … ►in which $2{\omega }_1$ and $2{\omega }_3$ are generators for the lattice $𝕃$, ${\omega }_2=-{\omega }_1-{\omega }_3$, and ${\eta }_j=\zeta \left({\omega }_j\right)$ (see (23.2.12)). …

24: 21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

25: 23.20 Mathematical Applications

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Rectangular Lattice

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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. …

26: 20.2 Definitions and Periodic Properties

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20.2.1 $${\theta }_1\left(z\right|\tau )={\theta }_1(z,q)=2\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n{q}^{{(n+\frac{1}{2})}^2}\sin \left((2n+1)z\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\sin z$: sine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.1

Referenced by:

§20.14, §20.2(i)

Permalink:

http://dlmf.nist.gov/20.2.E1

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Annotations for §20.2(i), §20.2 and Ch.20

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20.2.5 $$z_{m,n}=(m+n\tau )\pi ,$$ $m,n\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathbb{Z}$: set of all integers, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

A&S Ref:

16.33.1 (in different notation)

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.2.E5

Encodings:

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Annotations for §20.2(ii), §20.2 and Ch.20

►are the lattice points. The theta functions are quasi-periodic on the lattice: … ►

20.2.10 $$M\equiv M(z|\tau )={\mathrm{e}}^{\mathrm{i}z+(\mathrm{i}\pi \tau /4)},$$

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Defines:

$M$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter

A&S Ref:

16.33.1 (in different notation)

Permalink:

http://dlmf.nist.gov/20.2.E10

Encodings:

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See also:

Annotations for §20.2(iii), §20.2 and Ch.20

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27: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

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§26.3(i) Definitions

… ► $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\le n$, that stay on or above the line $y=x$ is $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►

Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

► ►►

$m$	$n$
	…

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…

28: 26.20 Physical Applications

… ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). …

29: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

… ►It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. …

30: 20.6 Power Series

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20.6.2 $${\theta }_1\left(\pi z\right|\tau )=\pi z{\theta }_1^{\prime }\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\delta }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\delta }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E2

Encodings:

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See also:

Annotations for §20.6 and Ch.20

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20.6.3 $${\theta }_2\left(\pi z\right|\tau )={\theta }_2\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\alpha }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\alpha }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E3

Encodings:

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See also:

Annotations for §20.6 and Ch.20

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20.6.7 $${\alpha }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+n\tau )}^{-2j},$$

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Defines:

${\alpha }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.6.E7

Encodings:

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Annotations for §20.6 and Ch.20

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20.6.8 $${\beta }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+(n-\frac{1}{2})\tau )}^{-2j},$$

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Defines:

${\beta }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E8

Encodings:

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See also:

Annotations for §20.6 and Ch.20

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20.6.9 $${\gamma }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m+(n-\frac{1}{2})\tau )}^{-2j},$$

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Defines:

${\gamma }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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