In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces...

particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties...

Maass form, and a mock theta function is essentially a mock modular form of weight ⁠1/2⁠. The first examples of mock theta functions were described by Srinivasa...

Riemann–Siegel theta function is defined in terms of the gamma function as θ ( t ) = arg ⁡ ( Γ ( 1 4 + i t 2 ) ) − log ⁡ π 2 t {\displaystyle \theta (t)=\arg...

functions. Elliptic curve Schwarz–Christoffel mapping Carlson symmetric form Jacobi theta function Ramanujan theta function Dixon elliptic functions Abel...

variables. Theta function may also refer to: q-theta function, θ ( z ; q ) {\displaystyle \theta (z;q)} , a type of q-series Theta function of a lattice...

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\,.} The equation θ = sin − 1 ⁡ x {\displaystyle \theta =\sin ^{-1}x}...

In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series...

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...

{\displaystyle \theta } , the sine and cosine functions are denoted as sin ⁡ ( θ ) {\displaystyle \sin(\theta )} and cos ⁡ ( θ ) {\displaystyle \cos(\theta )} ....

In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm. One can associate to any...

)} . An example is the Jacobi theta function θ ( z , τ ) = ∑ n = − ∞ ∞ e 2 π i n z   +   π i n 2 τ {\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty...

= ( θ − θ ^ ) 2 , {\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},} the risk function becomes the mean squared error of the...

via Bayes' rule. The likelihood function, parameterized by a (possibly multivariate) parameter θ {\textstyle \theta } , is usually defined differently...

_{0}^{\pi /2}{\frac {1}{({\sqrt[{z}]{\sin \theta }}+{\sqrt[{z}]{\cos \theta }})^{2z}}}\,d\theta } The beta function can be written as an infinite sum B ( x...

In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows: θ c ( z , m ) = 2 π q ( m ) 1 / 4 m 1 / 4 K ( m...

such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired...

mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(τ...

+2b\cdot \cos \theta \sin \theta +c\cdot \sin ^{2}\theta )}},\\\sigma _{Y}^{2}&={\frac {1}{2(a\cdot \sin ^{2}\theta -2b\cdot \cos \theta \sin \theta +c\cdot...

Riemann–Siegel theta function 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...

(s)=\int _{0}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t,} in terms of Jacobi's theta function θ ( τ ) = ∑ n = − ∞ ∞ e π i...

\,\tau >0.} The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function 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...

the remaining factor can be regarded as a function of the spherical angular coordinates θ {\displaystyle \theta } and φ {\displaystyle \varphi } only, or...

as Lovász theta function and is commonly denoted by ϑ ( G ) {\displaystyle \vartheta (G)} , using a script form of the Greek letter theta to contrast...

{versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}} There are several related functions corresponding...

representation along with the residue theorem. A second proof uses a theta function identity, or equivalently Poisson summation. These proofs are analogous...

comparison, the generating function of the regular partition numbers p(n) has this identity with respect to the theta function: ∑ n = 0 ∞ p ( n ) x n =...

elliptic functions Lemniscate elliptic functions Theta functions Neville theta functions Modular lambda function Closely related are the modular forms...