# with characteristics

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##### 1: 21.2 Definitions
###### §21.2(ii) Riemann Theta Functions with Characteristics
This function is referred to as a Riemann theta function with characteristics $\begin{bmatrix}\boldsymbol{{\alpha}}\\ \boldsymbol{{\beta}}\end{bmatrix}$. …
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. …
##### 2: 21.8 Abelian Functions
###### §21.8 Abelian Functions
For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …
##### 3: 28.34 Methods of Computation
###### §28.34(ii) Eigenvalues
• (f)

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

• ##### 4: 28.7 Analytic Continuation of Eigenvalues
###### §28.7 Analytic Continuation of Eigenvalues
The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …
28.7.4 $\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2n+2)^{2}\right)=0.$
##### 5: 28.17 Stability as $x\to\pm\infty$
The boundary of each region comprises the characteristic curves $a=a_{n}\left(q\right)$ and $a=b_{n}\left(q\right)$; compare Figure 28.2.1. …
##### 6: 21.3 Symmetry and Quasi-Periodicity
###### §21.3(ii) Riemann Theta Functions with Characteristics
21.3.4 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right)=e^{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}\theta% \genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(% \mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$
…For Riemann theta functions with half-period characteristics,
21.3.6 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=(-1)^{4\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\beta}}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{% \alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).$
##### 7: 28.30 Expansions in Series of Eigenfunctions
Let $\widehat{\lambda}_{m}$, $m=0,1,2,\dots$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_{m}(x)$, $m=0,1,2,\dots$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus
##### 9: 28.15 Expansions for Small $q$
###### §28.15(i) Eigenvalues $\lambda_{\nu}\left(q\right)$
28.15.2 $a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.$
##### 10: 21.1 Special Notation
Uppercase boldface letters are $g\times g$ real or complex matrices. The main functions treated in this chapter are the Riemann theta functions $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, and the Riemann theta functions with characteristics $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$. The function $\Theta(\boldsymbol{{\phi}}|\mathbf{B})=\theta\left(\boldsymbol{{\phi}}/(2\pi i% )\middle|\mathbf{B}/(2\pi i)\right)$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).