# theta functions

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##### 1: 20.2 Definitions and Periodic Properties
###### §20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
##### 2: 20.11 Generalizations and Analogs
###### §20.11(v) Permutation Symmetry
For $m=1,2,3,4$, $n=1,2,3,4$, and $m\neq n$, define twelve combined theta functions $\varphi_{m,n}\left(z,q\right)$ by …
##### 3: 27.13 Functions
Jacobi (1829) notes that $r_{2}\left(n\right)$ is the coefficient of $x^{n}$ in the square of the theta function $\vartheta\left(x\right)$:
27.13.4 $\vartheta\left(x\right)=1+2\sum_{m=1}^{\infty}x^{m^{2}},$ $|x|<1$.
(In §20.2(i), $\vartheta\left(x\right)$ is denoted by $\theta_{3}\left(0,x\right)$.) … Mordell (1917) notes that $r_{k}\left(n\right)$ is the coefficient of $x^{n}$ in the power-series expansion of the $k$th power of the series for $\vartheta\left(x\right)$. …
##### 4: 9.8 Modulus and Phase
9.8.4 $\theta\left(x\right)=\operatorname{arctan}\left(\mathrm{Ai}\left(x\right)/% \mathrm{Bi}\left(x\right)\right).$
(These definitions of $\theta\left(x\right)$ and $\phi\left(x\right)$ differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) … As $x$ increases from $-\infty$ to $0$ each of the functions $M\left(x\right)$, $M'\left(x\right)$, $|x|^{-1/4}N\left(x\right)$, $M\left(x\right)N\left(x\right)$, $\theta'\left(x\right)$, $\phi'\left(x\right)$ is increasing, and each of the functions $|x|^{1/4}M\left(x\right)$, $\theta\left(x\right)$, $\phi\left(x\right)$ is decreasing. …
##### 5: 21.2 Definitions
###### §21.2(i) Riemann ThetaFunctions
21.2.1 $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in% {\mathbb{Z}^{g}}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}% }\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}.$
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(ii) Riemann ThetaFunctions with Characteristics
21.2.5 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in{% \mathbb{Z}^{g}}}e^{2\pi i\left(\frac{1}{2}[\mathbf{n}+\boldsymbol{{\alpha}}]% \cdot\boldsymbol{{\Omega}}\cdot[\mathbf{n}+\boldsymbol{{\alpha}}]+[\mathbf{n}+% \boldsymbol{{\alpha}}]\cdot[\mathbf{z}+\boldsymbol{{\beta}}]\right)}.$
##### 6: 10.18 Modulus and Phase Functions
where $M_{\nu}\left(x\right)$ $(>0)$, $N_{\nu}\left(x\right)$ $(>0)$, $\theta_{\nu}\left(x\right)$, and $\phi_{\nu}\left(x\right)$ are continuous real functions of $\nu$ and $x$, with the branches of $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ fixed by …
10.18.9 ${N_{\nu}^{2}}\left(x\right)={M_{\nu}'^{2}}\left(x\right)+{M_{\nu}^{2}}\left(x% \right){\theta_{\nu}'^{2}}\left(x\right)={M_{\nu}'^{2}}\left(x\right)+\frac{4}% {(\pi xM_{\nu}\left(x\right))^{2}},$
10.18.11 $\tan\left(\phi_{\nu}\left(x\right)-\theta_{\nu}\left(x\right)\right)=\frac{M_{% \nu}\left(x\right)\theta_{\nu}'\left(x\right)}{M_{\nu}'\left(x\right)}=\frac{2% }{\pi xM_{\nu}\left(x\right)M_{\nu}'\left(x\right)},$
10.18.12 $M_{\nu}\left(x\right)N_{\nu}\left(x\right)\sin\left(\phi_{\nu}\left(x\right)-% \theta_{\nu}\left(x\right)\right)=\frac{2}{\pi x}.$
##### 7: 33.2 Definitions and Basic Properties
33.2.8 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)=e^{\pm\mathrm{i}{\theta_{\ell}}\left(% \eta,\rho\right)}(\mp 2\mathrm{i}\rho)^{\ell+1\pm\mathrm{i}\eta}U\left(\ell+1% \pm\mathrm{i}\eta,2\ell+2,\mp 2\mathrm{i}\rho\right),$
33.2.9 ${\theta_{\ell}}\left(\eta,\rho\right)=\rho-\eta\ln\left(2\rho\right)-\tfrac{1}% {2}\ell\pi+{\sigma_{\ell}}\left(\eta\right),$
##### 9: 21.9 Integrable Equations
###### §21.9 Integrable Equations
Typical examples of such equations are the Korteweg–de Vries equation …
##### 10: 21.10 Methods of Computation
###### §21.10(ii) Riemann ThetaFunctions Associated with a Riemann Surface
• Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.