lattice points

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1: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

►If ${\omega }_1$ and ${\omega }_3$ are nonzero real or complex numbers such that $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$, then the set of points $2m{\omega }_1+2n{\omega }_3$, with $m,n\in \mathbb{Z}$, constitutes a lattice $𝕃$ with $2{\omega }_1$ and $2{\omega }_3$ lattice generators. … ►The double series and double product are absolutely and uniformly convergent in compact sets in $ℂ$ that do not include lattice points. … ► $\mathrm{\wp }\left(z\right)$ and $\zeta \left(z\right)$ are meromorphic functions with poles at the lattice points. …The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. …

2: 20.2 Definitions and Periodic Properties

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

3: 23.9 Laurent and Other Power Series

… ►Let $z_0(\ne 0)$ be the nearest lattice point to the origin, and define …

4: Bibliography H

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Exponential sums and lattice points. III. Proc. London Math. Soc. (3) 87 (3), pp. 591–609.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (Angel V. Kumchev), ZentralBlatt

Referenced by:

§27.11, §27.11, Erratum (V1.1.8) for Section 27.11.

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5: 26.12 Plane Partitions

… ►It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. …

6: 23.20 Mathematical Applications

… ►There is a unique point $z_0\in [{\omega }_1,{\omega }_1+{\omega }_3]\cup [{\omega }_1+{\omega }_3,{\omega }_3]$ such that $\mathrm{\wp }\left(z_0\right)=0$. … ►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 $P=P(z)$. …

7: 23.6 Relations to Other Functions

… ►Let $z$ be a point of $ℂ$ different from $e_1,e_2,e_3$, and define $w$ by …

8: 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. …

9: 22.4 Periods, Poles, and Zeros

… ►The set of points $z=mK+n\mathrm{i}{K}^{\prime }$, $m,n\in \mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime }$, where again $m,n\in \mathbb{Z}$. …

10: 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)). …