translation by half-periods

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1: 20.2 Definitions and Periodic Properties

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§20.2(iii) Translation of the Argument by Half-Periods

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2: 22.4 Periods, Poles, and Zeros

§22.4 Periods, Poles, and Zeros

… ►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. …(b) The difference between p and the nearest q is a half-period of $\mathrm{p}\mathrm{q}(z,k)$. This half-period will be plus or minus a member of the triple $K,\mathrm{i}{K}^{\prime },K+\mathrm{i}{K}^{\prime }$; the other two members of this triple are quarter periods of $\mathrm{p}\mathrm{q}(z,k)$. ►

§22.4(iii) Translation by Half or Quarter Periods

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3: 21.2 Definitions

… ►It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor: … ►

21.2.7 $$\theta \left[\genfrac{}{}{0.0pt}{}{\mathrm{𝟎}}{\mathrm{𝟎}}\right]\left(𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $𝛀$: a Riemann matrix

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http://dlmf.nist.gov/21.2.E7

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

►Characteristics whose elements are either $0$ or $\frac{1}{2}$ are called half-period characteristics. For given $𝛀$, there are $2^{2g}$ $g$-dimensional Riemann theta functions with half-period characteristics. …