# theta functions

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## 11—20 of 174 matching pages

##### 11: 20.1 Special Notation
 $m$, $n$ integers. … $e^{i\alpha\pi\tau}$ for $\alpha\in\mathbb{R}$ (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 functions22.2); see Neville (1951). …
##### 13: 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. …
##### 14: 20.13 Physical Applications
###### §20.13 Physical Applications
with $\kappa=-i\pi/4$. … Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). … In the singular limit $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, 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.
##### 15: 20.4 Values at $z$ = 0
###### §20.4 Values at $z$ = 0
20.4.3 $\theta_{2}\left(0,q\right)=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}% \right)\left(1+q^{2n}\right)^{2},$
20.4.4 $\theta_{3}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+q^{2n-1}\right)^{2},$
20.4.5 $\theta_{4}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1-q^{2n-1}\right)^{2}.$
##### 16: 21.3 Symmetry and Quasi-Periodicity
###### §21.3(ii) Riemann ThetaFunctions with Characteristics
…For Riemann theta functions with half-period characteristics, …
##### 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. …
##### 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 $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, 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).