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1: 21.7 Riemann Surfaces

… ►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way. ►Since a Riemann surface $\mathrm{\Gamma }$ is a two-dimensional manifold that is orientable (owing to its analytic structure), its only topological invariant is its genus $g$ (the number of handles in the surface). …For example, Figure 21.7.1 depicts a genus 2 surface. … ►On a Riemann surface of genus $g$, there are $g$ linearly independent holomorphic differentials ${\omega }_j$, $j=1,2,\mathrm{\dots },g$. … ►The genus of this surface is $g$. …

2: 21.4 Graphics

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve ${\mu }^3+2\mu -{\lambda }^4=0$; compare §21.7(i). Magnify 3D Help

3: 21.2 Definitions

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21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

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Defines:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(i), §21.2 and Ch.21

… ► $\theta \left(𝐳\right|𝛀)$ is also referred to as a theta function with $g$ components, a $g$-dimensional theta function or as a genus $g$ theta function. …