theta functions

SearchAdvancedHelp

(0.010 seconds)

11—20 of 176 matching pages

11: 21.3 Symmetry and Quasi-Periodicity

… ►

§21.3(i) Riemann Theta Functions

… ► ►

§21.3(ii) Riemann Theta Functions with Characteristics

… ► …For Riemann theta functions with half-period characteristics, …

12: 20.1 Special Notation

… ► ►►►

$m$, $n$	integers.
…
$q^{\alpha }$	${\mathrm{e}}^{\mathrm{i}\alpha \pi \tau }$ for $\alpha \in ℝ$ (resolving issues of choice of branch).
…

… ►Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21. ►Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. … ►This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951). …

13: 20 Theta Functions

Chapter 20 Theta Functions

…

14: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►Such a solution is given in terms of a Riemann theta function with two phases. …

15: 20.13 Physical Applications

§20.13 Physical Applications

… ►with $\kappa =-\mathrm{i}\pi /4$. … ►Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). … ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

16: 20.4 Values at $z$ = 0

§20.4 Values at $z$ = 0

… ►

20.4.3 $${\theta }_2(0,q)=2q^{1/4}\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1+q^{2n}\right)}^2,$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Referenced by:

(20.10.3)

Permalink:

http://dlmf.nist.gov/20.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

►

20.4.4 $${\theta }_3(0,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1+q^{2n-1}\right)}^2,$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Referenced by:

(20.10.2)

Permalink:

http://dlmf.nist.gov/20.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

►

20.4.5 $${\theta }_4(0,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1-q^{2n-1}\right)}^2.$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Referenced by:

(20.10.1)

Permalink:

http://dlmf.nist.gov/20.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

►

Jacobi’s Identity

…

17: 20.3 Graphics

… ►

§20.3(i) $\theta$-Functions: Real Variable and Real Nome

… ►

►►

Figure 20.3.2: ${\theta }_1(\pi x,q)$, $0\le x\le 2$, $q$ = 0. … Magnify

… ►

►

►

Figure 20.3.13: ${\theta }_4(\pi x,q)$, $0\le x\le 2$, $0\le q\le 0.99$. Magnify 3D Help

►

§20.3(ii) $\theta$-Functions: Complex Variable and Real Nome

… ►

►

►

Figure 20.3.21: ${\theta }_4\left(0\right|u+\mathrm{i}v)$, $-1\le u\le 1$, $0.005\le v\le 0.1$. Magnify 3D Help

18: 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

…

19: 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 $\theta \left(𝐳\right|𝛀)$, and the Riemann theta functions with characteristics $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$. ►The function $\mathrm{\Theta }(\mathit{\varphi }|𝐁)=\theta \left(\mathit{\varphi }/(2\pi \mathrm{i})\right|𝐁/(2\pi \mathrm{i}))$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).

20: 20.7 Identities

… ►

§20.7(i) Sums of Squares

… ►

§20.7(ii) Addition Formulas

… ►

§20.7(v) Watson’s Identities

… ►

§20.7(vi) Landen Transformations

… ►

§20.7(vii) Derivatives of Ratios of Theta Functions

…