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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
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
Mathematical function
functions. Elliptic curve Schwarz–Christoffel mapping Carlson symmetric form Jacobi theta function Ramanujan theta function Dixon elliptic functions Abel
Jacobi_elliptic_functions
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
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
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)
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
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
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
Analytic function in mathematics
Arithmetic zeta function Generalized Riemann hypothesis Lehmer pair Prime zeta function Renormalization Riemann–Siegel theta function ZetaGrid "Jupyter
Riemann_zeta_function
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
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
Mathematical relation assigning a probability event to a cost
P θ {\displaystyle P_{\theta }} , of the observed data, X {\displaystyle X} . This is also referred to as the risk function of the decision rule δ {\displaystyle
Loss_function
Number, approximately 3.14
subgroups), a lattice in the group SL2(ℝ). An example is the Jacobi theta function θ(z,τ)=∑n=−∞∞e2πinz + πin2τ, which is a kind of modular form called
Pi
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
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
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
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
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
Special mathematical functions defined on the surface of a sphere
the remaining factor can be regarded as a function of the spherical angular coordinates θ {\displaystyle \theta } and φ {\displaystyle \varphi } only, or
Spherical_harmonics
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
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
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
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
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:
Beta_function
Antiderivative of the secant function
\sec \theta \,d\theta ={\begin{cases}{\dfrac {1}{2}}\ln {\dfrac {1+\sin \theta }{1-\sin \theta }}+C\\[15mu]\ln {{\bigl |}\sec \theta +\tan \theta \,{\bigr
Integral of the secant function
Integral_of_the_secant_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
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
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
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
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
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
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
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
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
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
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
Elliptic analog of hypergeometric series
\tau ]={\frac {\theta _{1}(\pi \sigma a,e^{\pi i\tau })}{\theta _{1}(\pi \sigma ,e^{\pi i\tau })}}} where the Jacobi theta function is defined by θ 1
Elliptic hypergeometric series
Elliptic_hypergeometric_series
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
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
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
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
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
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
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
{\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
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
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)
Gradient of the likelihood function
{L}}(\theta ;x)} , the natural logarithm of the likelihood function, with respect to an m-dimensional parameter vector θ {\displaystyle \theta } . s (
Informant_function
Indicator function of positive numbers
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside
Heaviside_step_function
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
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
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
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
Mathematical process of finding the derivative of a trigonometric function
}{\operatorname {d} \!\theta }}\,\sin \theta =(1)\cos \theta +(0)\sin \theta =\cos \theta \,.} We again calculate the derivative of the cosine function from the limit
Differentiation of trigonometric functions
Differentiation_of_trigonometric_functions
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
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
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
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
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
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
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
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
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
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)
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
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
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
Circle with radius of one
{\displaystyle \theta } from the positive real axis using the complex exponential function, z = e i θ = cos θ + i sin θ . {\displaystyle z=e^{i\theta }=\cos
Unit_circle
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
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
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
Geometric line segment whose endpoints lie on a circular arc
{\displaystyle \operatorname {crd} \theta ={\sqrt {(1-\cos \theta )^{2}+\sin ^{2}\theta }}={\sqrt {2-2\cos \theta }}=2\sin {\frac {\theta }{2}}} The last step uses
Chord_(geometry)
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
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
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
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)
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
Polynomial function of degree two
In mathematics, a quadratic function of a single variable is a function of the form f ( x ) = a x 2 + b x + c {\displaystyle f(x)=ax^{2}+bx+c} with
Quadratic_function
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
Fay's trisecant identity Fay, John D. (1973), "The prime- form", Theta functions on Riemann surfaces, Lecture Notes in Mathematics, vol. 352, Berlin
Prime_form
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
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
Class of statistical models
{\displaystyle {\boldsymbol {\theta }}} and τ {\displaystyle \tau } , whose density functions f (or probability mass function, for the case of a discrete
Generalized_linear_model
Paradigm for the design, analysis, and scoring of tests
{\displaystyle {\theta }} ; Local independence of items; The response of a person to an item can be modeled by a mathematical item response function (IRF). The
Item_response_theory
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
Trigonometric function defined as secant minus one
conjunction with the existing versine function, vers θ = 1 − cos θ {\displaystyle \operatorname {vers} \theta =1-\cos \theta } , for designing and measuring
Exsecant
Class of mathematical functions
{1}{2\pi }}\int _{0}^{2\pi }\varphi (z+re^{i\theta })\,d\theta .} Intuitively, this means that a subharmonic function is at any point no greater than the average
Subharmonic_function
ISSN 1340-6116, MR 2464529 Mochizuki, Shinichi (2009), "The étale theta function and its Frobenioid-theoretic manifestations", Kyoto University. Research
Frobenioid
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
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Conjecture on zeros of the zeta function
function has opposite signs at these points. Usually one writes ζ(12+it)=Z(t)e−iθ(t) where Hardy's Z function and the Riemann–Siegel theta function θ
Riemann_hypothesis
Topics referred to by the same term
from the reals onto α Theta (gastropod), a genus of sea snails Theta functions, special functions of several complex variables Theta meson, a hypothetical
Theta_(disambiguation)
Topics referred to by the same term
mysterious connection between Niemeier lattices and Ramanujan's mock theta functions Francisco Umbral (1932–2007), Spanish journalist, novelist, biographer
Umbral
System of complete and orthogonal polynomials
\theta )&=1&&=P_{0}(\cos \theta ),\\[4pt]T_{1}(\cos \theta )&=\cos \theta &&=P_{1}(\cos \theta ),\\[4pt]T_{2}(\cos \theta )&=\cos 2\theta &&={\tfrac
Legendre_polynomials
Mathematical concept
result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be
Infinite_product
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)
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
travel, tourism, insurance
THETA FUNCTION
THETA FUNCTION
Girl/Female
Hindu, Indian, Kannada, Sindhi
Innocent Beauty
Boy/Male
Hindu, Indian
Lighting
Girl/Female
Finnish, German, Gujarati, Hindu, Indian, Kannada
Love; Battle
Female
English
English variant spelling of Spanish Rita, RHETA means "pearl."Â
Female
Greek
(ΘήÏα) Greek name THERA means "lustrous." In mythology, this is the name of one of Amphion's seven daughters. Compare with another form of Thera.
Female
Greek
 Short form of Greek and Latin Dorothea, THEA means "gift of God." Compare with another form of Thea.
Girl/Female
Greek
Untamed.
Girl/Female
Hindu
Gift of God
Girl/Female
American, Australian, British, Christian, English, German, Greek
Gift of God; Supreme Gift
Girl/Female
American, Australian, Christian, Greek
Speaker; Pearl; Variant Form of Rita
Girl/Female
Greek
Speaker.
Boy/Male
Hindu, Indian
Quick
Female
English
 Pet form of English Theodora, THEA means "gift of God." Compare with another form of Thea.
Girl/Female
Egyptian
Queen.
Boy/Male
Hindu
Radiant
Female
English
Pet form of English Theodora, THEDA means "gift of God."
Girl/Female
Indian
Love
Girl/Female
Greek American
Goddess; godly. Also as abbreviation of names like Althea and Dorothea. The mythological Thea was...
Girl/Female
Russian American Greek
God's gift.
Female
Spanish
 Pet form of Spanish Theresa, THERA means "harvester." Compare with another form of Thera.
THETA FUNCTION
THETA FUNCTION
THETA FUNCTION
THETA FUNCTION
THETA FUNCTION
THETA FUNCTION
THETA FUNCTION
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