For other θ functions, see Theta function (disambiguation).
Jacobi's theta function θ1 with nome q = eiπτ = 0.1e0.1iπ:
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.[1]
The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent.
One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".[2]
Throughout this article, should be interpreted as (in order to resolve issues of choice of branch).[note 1]
^Tyurin, Andrey N. (30 October 2002). "Quantization, Classical and Quantum Field Theory and Theta-Functions". arXiv:math/0210466v1.
^Chang, Der-Chen (2011). Heat Kernels for Elliptic and Sub-elliptic Operators. Birkhäuser. p. 7.
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In mathematics, thetafunctions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces...
particularly q-analog theory, the Ramanujan thetafunction generalizes the form of the Jacobi thetafunctions, while capturing their general properties...
statistical model. Intuitively, the likelihood function L ( θ ∣ x ) {\displaystyle {\mathcal {L}}(\theta \mid x)} is the probability of observing data...
Maass form, and a mock thetafunction is essentially a mock modular form of weight 1/2. The first examples of mock thetafunctions were described by Srinivasa...
Theta (UK: /ˈθiːtə/, US: /ˈθeɪtə/; uppercase Θ or ϴ; lowercase θ or ϑ; Ancient Greek: θῆτα thē̂ta [tʰɛ̂ːta]; Modern: θήτα thī́ta [ˈθita]) is the eighth...
trigonometric function alternatively written arcsin x : {\displaystyle \arcsin x\colon } The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x}...
{\displaystyle \theta } , the sine and cosine functions are denoted as sin θ {\displaystyle \sin \theta } and cos θ {\displaystyle \cos \theta } . The definitions...
In mathematics, the Neville thetafunctions, named after Eric Harold Neville, are defined as follows: θ c ( z , m ) = 2 π q ( m ) 1 / 4 m 1 / 4 K ( m...
{R} )} . An example is the Jacobi thetafunction θ ( z , τ ) = ∑ n = − ∞ ∞ e 2 π i n z + i π n 2 τ {\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty...
trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt...
zeta function Generalized Riemann hypothesis Lehmer pair Prime zeta function Riemann Xi function Renormalization Riemann–Siegel thetafunction ZetaGrid...
f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.} Here Γ(k) is the gamma function evaluated...
such as the Ramanujan prime, the Ramanujan thetafunction, partition formulae and mock thetafunctions, have opened entire new areas of work and inspired...
required the introduction of the hyperelliptic thetafunction and later the general Riemann thetafunction for algebraic curves of arbitrary genus. The...
Riemann–Siegel thetafunction and the Riemann zeta function by Z ( t ) = e i θ ( t ) ζ ( 1 2 + i t ) . {\displaystyle Z(t)=e^{i\theta (t)}\zeta \left({\frac...
In mathematics, the thetafunction of a lattice is a function whose coefficients give the number of vectors of a given norm. One can associate to any...
\,\tau >0.} The thetafunction of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the thetafunction of an even unimodular...
{\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}.} The following improper integral involves the (not normalized) sinc function: ∫ 0 ∞ d x...
representation along with the residue theorem. A second proof uses a thetafunction identity, or equivalently Poisson summation. These proofs are analogous...
likelihood function over the parameter space, that is θ ^ = a r g m a x θ ∈ Θ L n ( θ ; y ) . {\displaystyle {\hat {\theta }}={\underset {\theta \in \Theta }{\operatorname...
}(-1)^{n}q^{\frac {3n^{2}-n}{2}}.} The Jacobi Triple Product also allows the Jacobi thetafunction to be written as an infinite product as follows: Let x = e i π τ {\displaystyle...
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