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1: 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. …

2: 21.3 Symmetry and Quasi-Periodicity

… ►The set of points ${𝐦}_1+𝛀{𝐦}_2$ form a $g$-dimensional lattice, the period lattice of the Riemann theta function. …

3: 23.7 Quarter Periods

§23.7 Quarter Periods

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23.7.1 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_1\right)=e_1+\sqrt{(e_1-e_3)(e_1-e_2)}=e_1+{\omega }_1^{-2}{(K\left(k\right))}^2{k}^{\prime },$$

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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, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

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23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

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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, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

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23.7.3 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_3\right)=e_3-\sqrt{(e_1-e_3)(e_2-e_3)}=e_3-{\omega }_1^{-2}{(K\left(k\right))}^{2}k,$$

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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, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

►where $k,k^{\prime }$ and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.

4: 23.2 Definitions and Periodic Properties

… ►Hence $\mathrm{\wp }\left(z\right)$ is an elliptic function, that is, $\mathrm{\wp }\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\mathrm{\wp }\left(z\right)$ is meromorphic and has two periods whose ratio is not real. … ►The function $\zeta \left(z\right)$ is quasi-periodic: for $j=1,2,3$, … ►For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).

5: 20.2 Definitions and Periodic Properties

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

6: 22.4 Periods, Poles, and Zeros

§22.4 Periods, Poles, and Zeros

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§22.4(i) Distribution

… ►Table 22.4.2 displays the periods and zeros of the functions in the $z$-plane in a similar manner to Table 22.4.1. … ►Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. … ►

§22.4(iii) Translation by Half or Quarter Periods

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7: 31.2 Differential Equations

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§31.2(iv) Doubly-Periodic Forms

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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$e_1+e_2+e_3=0,$

►where $2{\omega }_1$ and $2{\omega }_3$ with $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$ are generators of the lattice $𝕃$ for $\mathrm{\wp }\left(z|𝕃\right)$. …

8: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation ►

20.13.1 $$\partial \theta (z|\tau )/\partial \tau =\kappa {\partial }^2\theta (z|\tau )/{\partial z}^2,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\tau$: lattice parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

… ►Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). …

9: 23.6 Relations to Other Functions

… ►In this subsection $2{\omega }_1$, $2{\omega }_3$ are any pair of generators of the lattice $𝕃$, and the lattice roots $e_1$, $e_2$, $e_3$ are given by (23.3.9). … ►For further results for the $\sigma$-function see Lawden (1989, §6.2). … ►Again, in Equations (23.6.16)–(23.6.26), $2{\omega }_1,2{\omega }_3$ are any pair of generators of the lattice $𝕃$ and $e_1,e_2,e_3$ are given by (23.3.9). … ►

Rectangular Lattice

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General Lattice

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