theta functions

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21: 23.15 Definitions
§23.15 Definitions
23.15.6 $\lambda\left(\tau\right)=\frac{{\theta_{2}}^{4}\left(0,q\right)}{{\theta_{3}}^% {4}\left(0,q\right)};$
23.15.7 $J\left(\tau\right)=\frac{\left({\theta_{2}}^{8}\left(0,q\right)+{\theta_{3}}^{% 8}\left(0,q\right)+{\theta_{4}}^{8}\left(0,q\right)\right)^{3}}{54\left(\theta% _{1}'\left(0,q\right)\right)^{8}},$
23.15.9 $\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)\right)^{1/% 3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle|3\tau\right).$
22: Bernard Deconinck
He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. …
• 23: 21.5 Modular Transformations
§21.5(i) Riemann ThetaFunctions
Equation (21.5.4) is the modular transformation property for Riemann theta functions. The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi(\boldsymbol{{\Gamma}})$ is determinate: …
§21.5(ii) Riemann ThetaFunctions with Characteristics
For explicit results in the case $g=1$, see §20.7(viii).
24: 22.2 Definitions
where $k^{\prime}=\sqrt{1-k^{2}}$ and the theta functions are defined in §20.2(i). …
22.2.7 $\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}}^{2}\left(0,q\right)}{% \theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(% \zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}% \left(z,k\right)},$
22.2.9 $\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.$
The six functions containing the letter $\mathrm{s}$ in their two-letter name are odd in $z$; the other six are even in $z$. In terms of Neville’s theta functions20.1) …
26: 20.15 Tables
§20.15 Tables
Theta functions are tabulated in Jahnke and Emde (1945, p. 45). …
20.15.1 $\sin\alpha={\theta_{2}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)=k.$
Tables of Neville’s theta functions $\theta_{s}\left(x,q\right)$, $\theta_{c}\left(x,q\right)$, $\theta_{d}\left(x,q\right)$, $\theta_{n}\left(x,q\right)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\varepsilon,\alpha=0(5^{\circ})90^{\circ}$, where (in radian measure) $\varepsilon=x/{\theta_{3}}^{2}\left(0,q\right)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …
27: 20.10 Integrals
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). … For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).
28: 21.4 Graphics
§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 Figure 21.4.4: A real-valued scaled Riemann theta function: θ ^ ⁡ ( i ⁢ x , i ⁢ y | Ω 1 ) , 0 ≤ x ≤ 4 , 0 ≤ y ≤ 4 . … Magnify 3D Help Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: ℜ ⁡ θ ^ ⁡ ( x + i ⁢ y , 0 , 0 | Ω 2 ) , 0 ≤ x ≤ 1 , 0 ≤ y ≤ 3 . … Magnify 3D Help
29: 21.7 Riemann Surfaces
§21.7(i) Connection of Riemann ThetaFunctions to Riemann Surfaces
In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … is a Riemann matrix and it is used to define the corresponding Riemann theta function. …
30: 19.10 Relations to Other Functions
§19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …