with characteristics

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11: 28.36 Software

… ►

§28.36(ii) Characteristic Exponents and Eigenvalues

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12: 2.9 Difference Equations

… ►

§2.9(i) Distinct Characteristic Values

… ►where

\rho_{1},\rho_{2}

are the roots of the characteristic equation … ►

§2.9(ii) Coincident Characteristic Values

…

13: 21.6 Products

… ►On using theta functions with characteristics, it becomes ►

21.6.4

\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\sum_{k=1}^{h}T_{jk}\mathbf{c}_{k% }}{\sum_{k=1}^{h}T_{jk}\mathbf{d}_{k}}\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}% \middle|\boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}% \in\mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{-2\pi i\sum_{j=1}^{h}\mathbf{% b}_{j}\cdot\mathbf{c}_{j}}\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\mathbf% {a}_{j}+\mathbf{c}_{j}}{\mathbf{b}_{j}+\mathbf{d}_{j}}\left(\mathbf{z}_{j}% \middle|\boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $g$ : positive integer, $h$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $T_{jk}$ : element, $\mathcal{K}$ : set of matrices and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6 and Ch.21

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21.6.7

\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}+% \mathbf{c}_{3}+\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}+% \mathbf{d}_{3}+\mathbf{d}_{4}]}\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+% \mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{% }{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}-\mathbf{c}_{3}-\mathbf{c}_{4}]}{% \tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}-\mathbf{d}_{3}-\mathbf{d}_{4}]}% \left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol% {{\Omega}}\right)\*\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-% \mathbf{c}_{2}+\mathbf{c}_{3}-\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-% \mathbf{d}_{2}+\mathbf{d}_{3}-\mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y% }+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{% [}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-\mathbf{c}_{2}-\mathbf{c}_{3}+% \mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-\mathbf{d}_{2}-\mathbf{d}_{3}+% \mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}% \middle|\boldsymbol{{\Omega}}\right)\\ =\frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{% \mathbb{Z}}^{g}}\,\sum_{\boldsymbol{{\beta}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{% \mathbb{Z}}^{g}}e^{-2\pi i\boldsymbol{{\beta}}\cdot[\mathbf{c}_{1}+\mathbf{c}_% {2}+\mathbf{c}_{3}+\mathbf{c}_{4}]}\*\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {1}+\boldsymbol{{\alpha}}}{\mathbf{d}_{1}+\boldsymbol{{\beta}}}\left(\mathbf{x% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {2}+\boldsymbol{{\alpha}}}{\mathbf{d}_{2}+\boldsymbol{{\beta}}}\left(\mathbf{y% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {3}+\boldsymbol{{\alpha}}}{\mathbf{d}_{3}+\boldsymbol{{\beta}}}\left(\mathbf{u% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {4}+\boldsymbol{{\alpha}}}{\mathbf{d}_{4}+\boldsymbol{{\beta}}}\left(\mathbf{v% }\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

►

§21.6(ii) Addition Formulas

… ►

21.6.8

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\gamma}}}% \left(\mathbf{z}_{1}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0% .0pt}{}{\boldsymbol{{\beta}}}{\boldsymbol{{\delta}}}\left(\mathbf{z}_{2}% \middle|\boldsymbol{{\Omega}}\right)=\sum_{\boldsymbol{{\nu}}\in{\mathbb{Z}}^{% g}/\left(2{\mathbb{Z}}^{g}\right)}\theta\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[% \boldsymbol{{\alpha}}+\boldsymbol{{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{% \gamma}}+\boldsymbol{{\delta}}}\left(\mathbf{z}_{1}+\mathbf{z}_{2}\middle|2% \boldsymbol{{\Omega}}\right)\*\theta\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[% \boldsymbol{{\alpha}}-\boldsymbol{{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{% \gamma}}-\boldsymbol{{\delta}}}\left(\mathbf{z}_{1}-\mathbf{z}_{2}\middle|2% \boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, $\mathbb{Z}$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $\boldsymbol{{\delta}}$ : $g$ -dimensional vector and $\boldsymbol{{\gamma}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(ii), §21.6 and Ch.21

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14: 28.12 Definitions and Basic Properties

… ►

§28.12(i) Eigenvalues $\lambda_{\nu+2n}\left(q\right)$

… ► … ►

28.12.2

\lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.12(i), §28.12 and Ch.28

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15: 28.29 Definitions and Basic Properties

… ►

§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

… ►

28.29.9

2\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\nu$ : complex parameter, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution, $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
Referenced by:: §28.29(ii), §28.29(ii), §28.29(ii)
Permalink:: http://dlmf.nist.gov/28.29.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §28.29(ii), §28.29 and Ch.28

►This is the characteristic equation of (28.29.1), and

\cos\left(\pi\nu\right)

is an entire function of

\lambda

. Given

\lambda

together with the condition (28.29.6), the solutions

\pm\nu

of (28.29.9) are the characteristic exponents of (28.29.1). … ►For a given

\nu

, the characteristic equation

\bigtriangleup(\lambda)-2\cos\left(\pi\nu\right)=0

has infinitely many roots

\lambda

. …

16: 3.2 Linear Algebra

… ►is called the characteristic polynomial of

\mathbf{A}

and its zeros are the eigenvalues of

\mathbf{A}

. The multiplicity of an eigenvalue is its multiplicity as a zero of the characteristic polynomial (§3.8(i)). … … ►Its characteristic polynomial can be obtained from the recursion ►

3.2.23

p_{k+1}(\lambda)=(\lambda-\alpha_{k+1})p_{k}(\lambda)-\beta_{k+1}^{2}p_{k-1}(% \lambda),

k=0,1,\dots,n-1

ⓘ

Defines:: $p_{k}(\lambda)$ : characteristic polynomial (locally)
Symbols:: $\alpha_{j}$ : matrix, $\beta_{j}$ : matrix and $\lambda$ : eigenvalue
Referenced by:: §3.2(vi), §3.2(vi), Erratum (V1.1.3) for Subsection 3.2(vi)
Permalink:: http://dlmf.nist.gov/3.2.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §3.2(vi), §3.2 and Ch.3

…

17: 20.11 Generalizations and Analogs

… ►

§20.11(iv) Theta Functions with Characteristics

►Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …

18: 21.5 Modular Transformations

… ►

§21.5(ii) Riemann Theta Functions with Characteristics

►

21.5.9

\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{D}\boldsymbol{{\alpha}}-\mathbf{C}% \boldsymbol{{\beta}}+\tfrac{1}{2}\operatorname{diag}[\mathbf{C}\mathbf{D}^{% \mathrm{T}}]}{-\mathbf{B}\boldsymbol{{\alpha}}+\mathbf{A}\boldsymbol{{\beta}}+% \tfrac{1}{2}\operatorname{diag}[\mathbf{A}\mathbf{B}^{\mathrm{T}}]}\left(\left% [[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{\mathrm{T}}\mathbf{% z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][\mathbf{C}\boldsymbol{{% \Omega}}+\mathbf{D}]^{-1}\right)=\kappa(\boldsymbol{{\alpha}},\boldsymbol{{% \beta}},\boldsymbol{{\Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]}e^{\pi i\mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]^{-1}\mathbf{C}\right]\cdot\mathbf{z}}\theta\genfrac{[}{]}{0.0pt}{}% {\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $\operatorname{diag}\mathbf{A}$ : Transpose of $[A_{11},A_{22},\ldots,A_{gg}]$ and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(ii), §21.5 and Ch.21

…

19: 28.2 Definitions and Basic Properties

… ►This is the characteristic equation of Mathieu’s equation (28.2.1). … ►

§28.2(v) Eigenvalues $a_{n}$ , $b_{n}$

… ►

See accompanying text — Figure 28.2.1: Eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ of Mathieu’s equation as functions of $q$ for $0\leq q\leq 10$ , $n=0,1,2,3,4$ ( $a$ ’s), $n=1,2,3,4$ ( $b$ ’s). Magnify

►

Distribution

… ►

Change of Sign of $q$

…

20: 21.7 Riemann Surfaces

… ►

§21.7(ii) Fay’s Trisecant Identity

… ►

21.7.8

\zeta=\sum_{j=1}^{g}\omega_{j}\frac{\partial}{\partial z_{j}}\theta\genfrac{[}% {]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}% \middle|\boldsymbol{{\Omega}}\right)\Big{|}_{\mathbf{z}=\boldsymbol{{0}}}.

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector and $\omega$ : differential
Permalink:: http://dlmf.nist.gov/21.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.9

E(P_{1},P_{2})=\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{% \boldsymbol{{\beta}}}\left(\int_{P_{1}}^{P_{2}}\boldsymbol{{\omega}}\middle|% \boldsymbol{{\Omega}}\right)\Bigg{/}\left(\sqrt{\zeta(P_{1})}\sqrt{\zeta(P_{2}% )}\right),

ⓘ

Defines:: $E(\NVar{P_{1}},\NVar{P_{2}})$ : prime form (locally)
Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\int$ : integral, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $P_{1},P_{2}$ : points and $\boldsymbol{{\omega}}$ : vector of differentials
Permalink:: http://dlmf.nist.gov/21.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

… ► ►

§21.7(iii) Frobenius’ Identity

…