theta%0Afunctions

(0.003 seconds)

11—20 of 717 matching pages

11: 20.12 Mathematical Applications

§20.12 Mathematical Applications

►

§20.12(i) Number Theory

►For applications of

\theta_{3}\left(0,q\right)

to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). … ►Thus theta functions “uniformize” the complex torus. …

12: 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. … ► … ►

13: 21.6 Products

§21.6 Products

… ►Then …On using theta functions with characteristics, it becomes …Many identities involving products of theta functions can be established using these formulas. … ►For addition formulas for classical theta functions see §20.7(ii).

14: 20.9 Relations to Other Functions

… ►

§20.9(i) Elliptic Integrals

… ►and the notation of §19.2(ii), the complete Legendre integrals of the first kind may be expressed as theta functions: … ►

§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. … ►

§20.9(iii) Riemann Zeta Function

…

15: 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 …This Riemann matrix originates from the Riemann surface represented by the algebraic curve

\mu^{3}-\lambda^{7}+2\lambda^{3}\mu=0

; compare §21.7(i). … ►For the scaled Riemann theta functions depicted in Figures 21.4.2–21.4.5 … ►

See accompanying text — Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\Re\hat{\theta}\left(x+iy,0,0\middle|\boldsymbol{{\Omega}}_{2}\right)$ , $0\leq x\leq 1$ , $0\leq y\leq 3$ . … Magnify 3D Help

16: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

… ►

Jacobi’s Triple Product

… ► ►

§20.5(iii) Double Products

… ►

17: 21.5 Modular Transformations

… ►

§21.5(i) Riemann Theta Functions

… ►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 Theta Functions with Characteristics

… ►For explicit results in the case

g=1

, see §20.7(viii).

18: 19.11 Addition Theorems

… ►

\Delta(\theta)=\sqrt{1-k^{2}{\sin}^{2}\theta}.

… ►In the case of

\theta,\phi\in[0,\pi/2)

and

0\leq k^{2}\leq\alpha^{2}<\min\left(1,\left(1-\cos\theta\cos\phi\cos\psi\right% )^{-1}\right)

, we can use … ►If

\phi=\theta

in §19.11(i) and

\Delta(\theta)

is again defined by (19.11.3), then … ►

\cos\psi=(\cos\left(2\theta\right)+k^{2}{\sin}^{4}\theta)/(1-k^{2}{\sin}^{4}% \theta),

►

\tan\left(\tfrac{1}{2}\psi\right)=(\tan\theta)\Delta(\theta),

…

19: 22.2 Definitions

… ►

k=\frac{{\theta_{2}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q\right)},

… ►where

k^{\prime}=\sqrt{1-k^{2}}

and the theta functions are defined in §20.2(i). … ►For

k\in[0,1]

, all functions are real for

z\in\mathbb{R}

. … ►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 functions (§20.1) …

20: 21.1 Special Notation

… ► ►►►►

$g, h$	positive integers.
…
$\boldsymbol{{0}}_{g}$	$g\times g$ zero matrix.
…
$\mathbf{J}_{2g}$	$\begin{bmatrix}\boldsymbol{{0}}_{g}&\mathbf{I}_{g}\\ -\mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$ .
…

… ►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).