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Special functions of several complex variables
mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode
Theta_function
Mathematical function
particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties
Ramanujan_theta_function
Mathematical function
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
Riemann–Siegel_theta_function
Complex-differentiable part of a Maass wave function
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
Mock_modular_form
Upper bound on a graph's Shannon capacity
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
Lovász_number
Topics referred to by the same term
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 function (disambiguation)
Theta_function_(disambiguation)
Eighth letter of the Greek alphabet
Theta (uppercase Θ or ϴ; lowercase θ; cursive ϑ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth 𐤈. In the system
Theta
Mathematical function
^{2}\theta +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
Gaussian_function
Mathematical function
functions. Elliptic curve Schwarz–Christoffel mapping Carlson symmetric form Jacobi theta function Ramanujan theta function Dixon elliptic functions Abel
Jacobi_elliptic_functions
Functions of an angle
trigonometric function alternatively written arcsin x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x}
Trigonometric_functions
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
Neville_theta_functions
Fundamental trigonometric functions
{\displaystyle \theta } , the sine and cosine functions are denoted as sin ( θ ) {\displaystyle \sin(\theta )} and cos ( θ ) {\displaystyle \cos(\theta )} .
Sine_and_cosine
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
Q-theta_function
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
Theta_function_of_a_lattice
Inverse functions of sin, cos, tan, etc.
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
Inverse trigonometric functions
Inverse_trigonometric_functions
Function related to statistics and probability theory
{L}}(\theta \mid x)=p_{\theta }(x)=P_{\theta }(X=x)={\text{Pr}}\{X=x\mid \Theta =\theta \},} considered as a function of θ {\textstyle \theta } , a possible
Likelihood_function
Lattice in 8-dimensional space with special properties
\,\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
E8_lattice
Special mathematical functions defined on the surface of a sphere
_{m=-l}^{l}Y_{lm}^{*}(\theta \,',\,\phi \,')Y_{lm}(\theta ,\,\phi )=\delta (\phi -\phi \,')\,\delta (\cos \theta -\cos \theta \,').} The total power of a function f is
Spherical_harmonics
Mathematical relation assigning a probability event to a cost
{\displaystyle \theta } , and a quadratic loss function (squared error loss) L ( θ , θ ^ ) = ( θ − θ ^ ) 2 , {\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat
Loss_function
Number, approximately 3.14
. 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
Pi
Analytic function in mathematics
Particular values of the Riemann zeta function Prime zeta function Renormalization Riemann–Siegel theta function ZetaGrid "Jupyter Notebook Viewer". Nbviewer
Riemann_zeta_function
In mathematics, the Jacobi zeta function Z(u) is the logarithmic derivative of the Jacobi theta function Θ(u). It is also commonly denoted as zn ( u
Jacobi_zeta_function
Mathematical function
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
Z_function
Zeta-like functions approximate arbitrary holomorphic functions
g(s) can be approximated by a function of the form ln ( ζ M ( s , θ ^ ) ) {\displaystyle \ln(\zeta _{M}(s,{\hat {\theta }}))} for a suitable set M of
Zeta_function_universality
Indian mathematician (1887–1920)
such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired
Srinivasa_Ramanujan
formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes
Weil–Brezin_Map
Restriction of a theta function
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
Theta_constant
1 minus the cosine of an angle
{versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}} There are several related functions corresponding
Versine
Mathematical function
/2}{\frac {1}{\left({\sqrt[{z}]{\sin \theta }}+{\sqrt[{z}]{\cos \theta }}\right)^{2z}}}\,d\theta } The beta function can be written as an infinite sum B
Beta_function
Special mathematical function defined as sin(x)/x
{\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}.} The following improper integral involves the (not normalized) sinc function: ∫ 0 ∞ d x
Sinc_function
Mathematical function
Jacobi Theta function and ϑ 1 ( z | τ ) = − ϑ 11 ( z ; τ ) {\displaystyle \vartheta _{1}(z|\tau )=-\vartheta _{11}(z;\tau )} Because the eta function is easy
Dedekind_eta_function
Probability distribution
{\displaystyle X\sim \Gamma (\alpha ,\theta )\equiv \operatorname {Gamma} (\alpha ,\theta )} The probability density function using the shape-scale parametrization
Gamma_distribution
principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of A of dimension dim A − 1
Theta_divisor
Mathematic function
{\displaystyle \Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q)\,} where θ is the q-theta function. When p = 0 {\displaystyle p=0} , it essentially reduces to the infinite
Elliptic_gamma_function
connections and consistent notation with the Ramanujan G- and g-functions and the Jacobi theta functions, both of which conventionally uses the nome. Still employing
Weber_modular_function
Transcendental single-variable function
\operatorname {Sl} _{2m+1}(\theta )=\sum _{k=1}^{\infty }{\frac {\sin k\theta }{k^{2m+1}}}} N.B. The SL-type Clausen functions have the alternative notation
Clausen_function
quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg
Theta_representation
Special function in mathematics
representation along with the residue theorem. A second proof uses a theta function identity, or equivalently Poisson summation. These proofs are analogous
Hurwitz_zeta_function
Elliptic analog of hypergeometric series
modified Jacobi theta function with argument x and nome p is defined by θ ( x ; p ) = ( x , p / x ; p ) ∞ {\displaystyle \displaystyle \theta (x;p)=(x,p/x;p)_{\infty
Elliptic hypergeometric series
Elliptic_hypergeometric_series
Mathematical identity found by Jacobi in 1829
y^{2}=-q{\sqrt {q}}} . The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows: Let x = e i π τ {\displaystyle
Jacobi_triple_product
Mathematical functions
exponential function. An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see Lemniscate
Lemniscate_elliptic_functions
the Barnes–Wall lattice B W 16 {\displaystyle BW_{16}} . The lattice theta function for the Barnes Wall lattice B W 16 {\displaystyle BW_{16}} is known
Barnes–Wall_lattice
Arctangent function with two arguments
computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, θ = atan2 ( y , x ) {\displaystyle \theta =\operatorname {atan2}
Atan2
Technique in analytic number theory
theory of theta functions. In the context of Waring's problem, powers of theta functions are the generating functions for the sum of squares function. Their
Hardy–Ramanujan–Littlewood circle method
Hardy–Ramanujan–Littlewood_circle_method
Figurate number
the sum of triangular numbers are connected to theta functions, in particular the Ramanujan theta function. The number of line segments between closest
Triangular_number
Fourier transform of the probability density function
_{\mathbf {R} }g(t+\theta ){\overline {g(\theta )}}\,d\theta .} Mathias’ theorem. A real-valued, even, continuous, absolutely integrable function φ, with φ(0)
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
elliptic functions Lemniscate elliptic functions Theta functions Neville theta functions Modular lambda function Closely related are the modular forms
List of mathematical functions
List_of_mathematical_functions
Mathematical function, denoted exp(x) or e^x
θ {\displaystyle e^{i\theta }=\cos \theta +i\sin \theta } expresses and summarizes these relations. The exponential function can be even further generalized
Exponential_function
German mathematician (1804–1851)
required the introduction of the hyperelliptic theta function and later the general Riemann theta function for algebraic curves of arbitrary genus. The
Carl_Gustav_Jacob_Jacobi
{\begin{aligned}1+\cot ^{2}\theta &=\csc ^{2}\theta \\1+\tan ^{2}\theta &=\sec ^{2}\theta \\\sec ^{2}\theta +\csc ^{2}\theta &=\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}}
List of trigonometric identities
List_of_trigonometric_identities
mathematics, a Siegel theta series is a Siegel modular form associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice
Siegel_theta_series
Analytic function on the upper half-plane with a certain behavior under the modular group
mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of
Modular_form
Modular function in mathematics
lambda function λ ( τ ) = θ 2 ( e π i τ ) 4 θ 3 ( e π i τ ) 4 = k ( τ ) 2 {\displaystyle \lambda (\tau )={\frac {\theta _{2}(e^{\pi i\tau })^{4}}{\theta _{3}(e^{\pi
J-invariant
Number of partitions of an integer
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 =
Partition function (number theory)
Partition_function_(number_theory)
24-dimensional repeating pattern of points
{Im} \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
Leech_lattice
S-shaped curve
(\theta _{1},\theta _{2},\theta _{3})} is set to ( 10000 , 0.2 , 40 ) {\displaystyle (10000,0.2,40)} . One of the benefits of using a growth function such
Logistic_function
Mathematics award
of cusp points in the boundary of the Teichmüller space, and Kra's theta-function conjecture." 2002 Beijing, China Laurent Lafforgue Institut des hautes
Fields_Medal
Number-theoretical function
generating function of the sequence r k ( n ) {\displaystyle r_{k}(n)} for fixed k can be expressed in terms of the Jacobi theta function: ϑ ( 0 ; q )
Sum_of_squares_function
Mathematical constants
_{k=-\infty }^{\infty }{\frac {\theta _{4}(ik\pi ,e^{-\pi })}{e^{2\pi k^{2}}}},} where θ1 and θ4 are two of the Jacobi theta functions. There also exist a number
Particular values of the gamma function
Particular_values_of_the_gamma_function
Identity between theta functions of Riemann surfaces
between theta functions of Riemann surfaces introduced by John Fay. Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions
Fay's_trisecant_identity
notational systems for the Jacobi theta functions. The notations given in the Wikipedia article define the original function ϑ 00 ( z ; τ ) = ∑ n = − ∞ ∞ exp
Jacobi theta functions (notational variations)
Jacobi_theta_functions_(notational_variations)
Class of reinforcement learning algorithms
{\displaystyle \theta } . In policy-based RL, the actor is a parameterized policy function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are
Policy_gradient_method
Topics referred to by the same term
function Thomae's function, also called the Riemann function Riemann theta function, Riemann function, used in the Riemann method of solving the linear
Riemann_function
there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic
Theta_characteristic
Large countable ordinal
which acts as the limit of the range of Buchholz's psi function and Feferman's theta function. It was named by David Madore, after Gaisi Takeuti, Solomon
Takeuti–Feferman–Buchholz ordinal
Takeuti–Feferman–Buchholz_ordinal
Method of solution to differential equations
{x^{2}+y^{2}}}} , Θ ( t ) {\textstyle \Theta (t)} is the Heaviside step function, J ν ( z ) {\textstyle J_{\nu }(z)} is a Bessel function, I ν ( z ) {\textstyle I_{\nu
Green's_function
Group in mathematical representation theory
representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence. One way
Metaplectic_group
Conjecture on zeros of the zeta function
\zeta ({\tfrac {1}{2}}+it)=Z(t)e^{-i\theta (t)}} where Hardy's Z function and the Riemann–Siegel theta function θ are uniquely defined by this and the
Riemann_hypothesis
Class of mathematical functions
function ℘ ( z , τ ) = ℘ ( z , 1 , ω 2 / ω 1 ) {\displaystyle \wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})} can be represented by Jacobi's theta functions:
Weierstrass_elliptic_function
German mathematician (1826–1866)
Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated
Bernhard_Riemann
Method of estimating the parameters of a statistical model, given observations
{\displaystyle ~{\hat {\theta }}={\hat {\theta }}_{n}(\mathbf {y} )\in \Theta ~} that maximizes the likelihood function L n {\displaystyle \,{\mathcal
Maximum_likelihood_estimation
Class of functions behaving "like" periodic functions
if the function obeys the equation: f ( z + ω ) = C f ( z ) {\displaystyle f(z+\omega )=Cf(z)} An example of this is the Jacobi theta function, where
Quasiperiodic_function
Representation theory of the symplectic group
number theory, in particular to give a group theoretic explanation of theta functions and quadratic reciprocity. Several physicists and mathematicians observed
Oscillator_representation
Logarithm of a complex number
logarithm function along the unit circle, by evaluating L ( e i θ ) {\displaystyle \operatorname {L} \left(e^{i\theta }\right)} as θ {\displaystyle \theta }
Complex_logarithm
Complex exponential in terms of sine and cosine
θ ) {\displaystyle f(\theta )={\frac {\cos \theta +i\sin \theta }{e^{i\theta }}}=e^{-i\theta }\left(\cos \theta +i\sin \theta \right)} for real θ. Differentiating
Euler's_formula
Neural oscillatory pattern
Theta waves generate the theta rhythm, a neural oscillation in the brain that underlies various aspects of cognition and behavior, including learning,
Theta_wave
Generating function in integrable systems
{\text{Im}}(B){\text{ is positive definite}}\right\}.} The Riemann θ {\displaystyle \theta } function on C g {\displaystyle \mathbf {C} ^{g}} corresponding to the period
Tau function (integrable systems)
Tau_function_(integrable_systems)
Mathematical identities related to integer partitions
following identities to the remaining Rogers–Ramanujan functions and to the Ramanujan theta function described above: S ( q ) = q 1 / 5 H ( − q ) G ( − q
Rogers–Ramanujan_identities
Field of economics and game theory
y(\theta )=\{x(\theta ),t(\theta )\},\ x\in X,t\in T} where x {\displaystyle x} stands for an allocation of goods rendered or received as a function of
Mechanism_design
Polynomial sequence
{\max \left(|\cos \theta |^{-1},2\sin \theta \right)}{(2\sin \theta )^{M+\lambda }}}} where Γ {\displaystyle \Gamma } is the Gamma function. Other asymptotic
Gegenbauer_polynomials
Special mathematical function
description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents
Nome_(mathematics)
Special function defined by an integral
elliptic functions Jacobi theta function Meridian arc Pendulum period Ramanujan theta function Schwarz–Christoffel mapping Weierstrass's elliptic functions K
Elliptic_integral
Kind of complex manifold
varieties) when n > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing
Complex_torus
Function in fluid dynamics
{\displaystyle z=r\,\cos \theta \,} and ρ = r sin θ . {\displaystyle \rho =r\,\sin \theta .\,} As explained in the general stream function article, definitions
Stokes_stream_function
Quadratic homogeneous polynomial in two variables
reduced binary quadratic forms of a given discriminant. The classical theta function of 2 variables is ∑ ( m , n ) ∈ Z 2 q m 2 + n 2 {\displaystyle \sum
Binary_quadratic_form
Relation between sine and cosine
^{2}\theta +\sin ^{2}\theta &=\cos ^{2}\theta -i^{2}\sin ^{2}\theta \\[3mu]&=(\cos \theta +i\sin \theta )(\cos \theta -i\sin \theta )\\[3mu]&=e^{i\theta }e^{-i\theta
Pythagorean trigonometric identity
Pythagorean_trigonometric_identity
finite fields Riemann theta function Riemann Xi function Riemann zeta function Riemann–Siegel formula Riemann–Siegel theta function Free Riemann gas also
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
Topic in group theory and harmonic analysis (Niemeier lattice-mock theta connection)
mysterious connection between Niemeier lattices and Ramanujan's mock theta functions. It is a generalization of the Mathieu moonshine phenomenon connecting
Umbral_moonshine
Concept in probability theory
given a likelihood function p ( x ∣ θ ) {\displaystyle p(x\mid \theta )} , the posterior distribution p ( θ ∣ x ) {\displaystyle p(\theta \mid x)} is in the
Conjugate_prior
Measure of the error of an estimator
_{\theta }[{\hat {\theta }}]-\theta \right)+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta
Mean_squared_error
Group in group theory and physics
square integrable functions. In the theta, or holomorphic, model, the Heisenberg group acts on a Hilbert space of entire functions, with the model depending
Heisenberg_group
who exhibited the zeta function as the Mellin transform of Jacobi's theta function. Riemann used asymptotics of the theta function to obtain the analytic
Rankin–Selberg_method
Second-order partial differential equation
{\displaystyle \lambda \sin ^{2}\theta +{\frac {\sin \theta }{\Theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=m^{2}} for some
Laplace's_equation
Phrase in linguistics
Theta roles are the names of the participant roles associated with a predicate: the predicate may be a verb, an adjective, a preposition, or a noun. If
Theta_role
r=\operatorname {sn} (s,k),\,\theta =k\cdot s{\text{ and }}z=\operatorname {cn} (s,k)} or expressed as Jacobi theta functions r = ϑ 3 ( 0 ) ⋅ ϑ 1 ( s ⋅ ϑ
Seiffert's_spiral
Series representing modular forms
on modular invariants provides expressions for these two functions in terms of theta functions. Any holomorphic modular form for the modular group can
Eisenstein_series
Mathematical process of finding the derivative of a trigonometric function
\lim _{\theta \to 0^{-}}\!{\frac {\sin \theta }{\theta }}\ =\ \lim _{\theta \to 0^{+}}\!{\frac {\sin(-\theta )}{-\theta }}\ =\ \lim _{\theta \to 0^{+}}\
Differentiation of trigonometric functions
Differentiation_of_trigonometric_functions
ISSN 1340-6116, MR 2464529 Mochizuki, Shinichi (2009), "The étale theta function and its Frobenioid-theoretic manifestations", Kyoto University. Research
Frobenioid
Collection of Srinivasa Ramanujan's discoveries in mathematics
the most famous objects examined in the lost notebook are the mock theta functions ... Some time between 1934 and 1947, Hardy probably passed the notebook
Ramanujan's_lost_notebook
Equation in Fourier analysis
the theta function: θ ( τ ) = ∑ n q n 2 . {\displaystyle \theta (\tau )=\sum _{n}q^{n^{2}}.} The relation between θ ( − 1 / τ ) {\displaystyle \theta (-1/\tau
Poisson_summation_formula
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