# scaled Riemann theta functions

(0.002 seconds)

## 4 matching pages

##### 1: 21.4 Graphics
Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, with … For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5
##### 2: 21.2 Definitions
$\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is also referred to as a theta function with $g$ components, a $g$-dimensional theta function or as a genus $g$ theta function. For numerical purposes we use the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, defined by (Deconinck et al. (2004)),
21.2.2 $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{-\pi[\Im% \mathbf{z}]\cdot[\Im\boldsymbol{{\Omega}}]^{-1}\cdot[\Im\mathbf{z}]}\theta% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$
21.2.4 $\hat{\theta}\left(x_{1}+iy_{1},x_{2}+iy_{2}\middle|\begin{bmatrix}i&-\tfrac{1}% {2}\\ -\tfrac{1}{2}&i\end{bmatrix}\right)=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=-% \infty}^{\infty}e^{-\pi(n_{1}+y_{1})^{2}-\pi(n_{2}+y_{2})^{2}}\*e^{\pi i(2n_{1% }x_{1}+2n_{2}x_{2}-n_{1}n_{2})}.$
##### 3: 21.9 Integrable Equations
###### §21.9 Integrable Equations
Typical examples of such equations are the Korteweg–de Vries equation …Here, and in what follows, $x,y$, and $t$ suffixes indicate partial derivatives. …
##### 4: Errata
• Rearrangement

In previous versions of the DLMF, in §8.18(ii), the notation $\widetilde{\Gamma}(z)$ was used for the scaled gamma function $\Gamma^{*}\left(z\right)$. Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).

Equations: (5.9.2_5), (5.9.10_1), (5.9.10_2), (5.9.11_1), (5.9.11_2), the definition of the scaled gamma function $\Gamma^{*}\left(z\right)$ was inserted after the first equals sign in (5.11.3), post equality added in (7.17.2) which gives “$=\sum_{m=0}^{\infty}a_{m}t^{2m+1}$”, (7.17.2_5), (31.11.3_1), (31.11.3_2) with some explanatory text.

• Equation (19.25.37)

The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

• Equation (25.2.4)

The original constraint, $\Re s>0$, was removed because, as stated after (25.2.1), $\zeta\left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta\left(s\right)-(s-1)^{-1}$ is an entire function.

Suggested by John Harper.

• Equation (21.3.4)
21.3.4 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right)={\mathrm{e}}^{2\pi\mathrm{i}\boldsymbol{{\alpha}}\cdot\mathbf{% m}_{2}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta% }}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$

Originally the vector $\mathbf{m}_{2}$ on the right-hand side was given incorrectly as $\mathbf{m}_{1}$.

Reported 2012-08-27 by Klaas Vantournhout.