quasi-periodicity

SearchAdvancedHelp

(0.001 seconds)

6 matching pages

1: 21.3 Symmetry and Quasi-Periodicity

§21.3 Symmetry and Quasi-Periodicity

… ►

21.3.3 $$\theta \left(𝐳+{𝐦}_1+𝛀{𝐦}_2\right|𝛀)={\mathrm{e}}^{-2\pi \mathrm{i}\left(\frac{1}{2}{𝐦}_2\cdot 𝛀\cdot {𝐦}_2+{𝐦}_2\cdot 𝐳\right)}\theta \left(𝐳\right|𝛀),$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

… ►This is the quasi-periodicity property of the Riemann theta function. … ► …

2: 20.2 Definitions and Periodic Properties

… ►

§20.2(ii) Periodicity and Quasi-Periodicity

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

3: 22.16 Related Functions

… ►

Quasi-Periodicity

… ►

Quasi-Addition and Quasi-Periodic Formulas

…

4: 21.9 Integrable Equations

… ►The KP equation has a class of quasi-periodic solutions described by Riemann theta functions, given by …

5: 23.2 Definitions and Periodic Properties

… ►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).

6: 21.4 Graphics

… ►

►

►

Figure 21.4.4: A real-valued scaled Riemann theta function: $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|{𝛀}_1)$, $0\le x\le 4$, $0\le y\le 4$. In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration. Magnify 3D Help

…