# translation by half-periods

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## 3 matching pages

##### 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 $\operatorname{pq}\left(z,k\right)$. This half-period will be plus or minus a member of the triple ${K,iK^{\prime},K+iK^{\prime}}$; the other two members of this triple are quarter periods of $\operatorname{pq}\left(z,k\right)$.
##### 3: 21.2 Definitions
It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor: …
21.2.7 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{0}}}{\boldsymbol{{0}}}\left(\mathbf% {z}\middle|\boldsymbol{{\Omega}}\right)=\theta\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right).$
Characteristics whose elements are either $0$ or $\tfrac{1}{2}$ are called half-period characteristics. For given $\boldsymbol{{\Omega}}$, there are $2^{2g}$ $g$-dimensional Riemann theta functions with half-period characteristics. …