About the Project
NIST

theta functions

AdvancedHelp

(0.005 seconds)

11—20 of 174 matching pages

11: 20.1 Special Notation
m , n

integers.

q α

e i α π τ for α (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 θ symbols indicate derivatives with respect to the argument of the θ function. … This notation simplifies the relationship of the theta functions to Jacobian elliptic functions22.2); see Neville (1951). …
12: 20 Theta Functions
Chapter 20 Theta Functions
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 κ = - i π / 4 . … Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). … In the singular limit τ 0 + , the functions θ j ( z | τ ) , 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 θ 2 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) ( 1 + q 2 n ) 2 ,
20.4.4 θ 3 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 + q 2 n - 1 ) 2 ,
20.4.5 θ 4 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 - q 2 n - 1 ) 2 .
Jacobi’s Identity
16: 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, …
17: 20.3 Graphics
§20.3(i) θ -Functions: Real Variable and Real Nome
See accompanying text
Figure 20.3.2: θ 1 ( π x , q ) , 0 x 2 , q = 0. … Magnify
See accompanying text
Figure 20.3.13: θ 4 ( π x , q ) , 0 x 2 , 0 q 0.99 . Magnify 3D Help
§20.3(ii) θ -Functions: Complex Variable and Real Nome
See accompanying text
Figure 20.3.21: θ 4 ( 0 | u + i v ) , - 1 u 1 , 0.005 v 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 × g real or complex matrices. The main functions treated in this chapter are the Riemann theta functions θ ( z | Ω ) , and the Riemann theta functions with characteristics θ [ α β ] ( z | Ω ) . The function Θ ( ϕ | B ) = θ ( ϕ / ( 2 π i ) | B / ( 2 π 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