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Topics referred to by the same term
eta function may refer to: The Dirichlet eta function η(s), a Dirichlet series The Dedekind eta function η(τ), a modular form The Weierstrass eta function
Eta_function
Function in analytic number theory
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any
Dirichlet_eta_function
Mathematical function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane
Dedekind_eta_function
Seventh letter in the Greek alphabet
Eta (/ˈiːtə, ˈeɪtə/ EE-tə, AY-tə; uppercase Η, lowercase η; Ancient Greek: ἦτα ē̂ta [ɛ̂ːta] or Greek: ήτα ita [ˈita]) is the seventh letter of the Greek
Eta
Topics referred to by the same term
Weierstrass function may refer to: Weierstrass function (nowhere-differentiable function) Weierstrass sigma, zeta, and eta functions Weierstrass elliptic
Weierstrass_function
Mathematical functions related to Weierstrass's elliptic function
sigma, zeta, and eta functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for
Weierstrass sigma, zeta, and eta functions
Weierstrass_sigma,_zeta,_and_eta_functions
Conditions for switching order of integration in calculus
Dirichlet eta function as follows: η ( s ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n s = 1 − 1 2 s + 1 3 s − 1 4 s + 1 5 s − 1 6 s ± ⋯ {\displaystyle \eta (s)=\sum _{n=1}^{\infty
Fubini's_theorem
Evaluates a certain product of values of the Gamma function at rational values
certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers
Chowla–Selberg_formula
Mathematical identities related to integer partitions
eta function in their Weber form: G M ( q ) = η W ( q 5 ) 1 / 2 η W ( q ) − 1 / 2 R ( q ) − 1 / 2 {\displaystyle G_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta
Rogers–Ramanujan_identities
Special functions of several complex variables
1 , {\displaystyle \tau =n{\sqrt {-1}},} and Dedekind eta function η ( τ ) . {\displaystyle \eta (\tau ).} Then for n = 1 , 2 , 3 , … {\displaystyle n=1
Theta_function
Synchrotron function Riemann zeta function: A special case of Dirichlet series. Riemann Xi function Dirichlet eta function: An allied function. Dirichlet
List of mathematical functions
List_of_mathematical_functions
Mathematical theorem about the real analytic Eisenstein series
a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated
Kronecker_limit_formula
Differential operator
generalization of the Dirichlet eta function. They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional
Eta_invariant
Function studied by Ramanujan
(q)^{24}=\eta (z)^{24}=\Delta (z),} where ϕ {\displaystyle \phi } is the Euler function, η {\displaystyle \eta } is the Dedekind eta function, Δ ( z )
Ramanujan_tau_function
Polynomials used in approximation theory
height as the equiripple peaks. The Jacobi eta function can be defined in terms of a Jacobi auxiliary theta function, H ( φ | κ ) = θ 1 ( a | b ) {\displaystyle
Zolotarev_polynomials
Natural number
modular forms through the Dedekind eta function η ( τ ) = q 1 / 24 ∏ n > 0 ( 1 − q n ) , q = e 2 π i τ . {\displaystyle \eta (\tau )=q^{1/24}\prod _{n>0}(1-q^{n})
24_(number)
signature defect of the boundary of a manifold as the eta invariant, the value as s=0 of their eta function, and used this to show that Hirzebruch's signature
Shimizu_L-function
Topics referred to by the same term
Look up eta or ETA in Wiktionary, the free dictionary. Eta (Η or η) is the seventh letter of the Greek alphabet. Eta or ETA may also refer to: Eta (given
Eta_(disambiguation)
\eta (2\tau )}{\eta (\tau )}}.\end{aligned}}} These are also the definitions in Duke's paper "Continued Fractions and Modular Functions". The function
Weber_modular_function
Divergent series
uses the relationship between the Riemann zeta function and the Dirichlet eta function η(s). The eta function is defined by an alternating Dirichlet series
1_+_2_+_3_+_4_+_⋯
products of a sawtooth function. Dedekind introduced them in the 1880's to express the functional equation of the Dedekind eta function, in a commentary to
Dedekind_sum
Analytic function in mathematics
developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision
Riemann_zeta_function
Number of partitions of an integer
specifically the Dedekind eta function. The same sequence of pentagonal numbers appears in a recurrence relation for the partition function: p ( n ) = ∑ k ∈ Z
Partition function (number theory)
Partition_function_(number_theory)
Modular function in mathematics
)^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta (\tau )^{24}} , Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} , and modular invariants, g 2 (
J-invariant
Series related to Ramanujan's pi formulas
\end{aligned}}} with the j-function j(τ), Eisenstein series E4, and Dedekind eta function η(τ). The first expansion is the McKay–Thompson
Ramanujan–Sato_series
Topics referred to by the same term
function can refer to any of three functions, all introduced by Richard Dedekind Dedekind eta function Dedekind psi function Dedekind zeta function This
Dedekind_function
Conjecture on zeros of the zeta function
this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation (1−22s)ζ(s)=η(s)=∑n=1∞(−1)n+1ns=11s−12s+13s−⋯
Riemann_hypothesis
Sporadic simple group
{\eta (\tau )\,\eta (3\tau )}{\eta (2\tau )\,\eta (6\tau )}}\right)^{3}+2^{3}\left({\tfrac {\eta (2\tau )\,\eta (6\tau )}{\eta (\tau )\,\eta (3\tau
Fischer_group_Fi22
Class of mathematical functions
24 {\displaystyle \Delta =(2\pi )^{12}\eta ^{24}} where η {\displaystyle \eta } is the Dedekind eta function. For the Fourier coefficients of Δ {\displaystyle
Weierstrass_elliptic_function
Mathematical function
{\displaystyle (3n^{2}-n)/2} is a pentagonal number. The Euler function is related to the Dedekind eta function as ϕ ( e 2 π i τ ) = e − π i τ / 12 η ( τ ) . {\displaystyle
Euler_function
Analytic function on the upper half-plane with a certain behavior under the modular group
discriminant The Dedekind eta function is defined as η ( z ) = q 1 / 24 ∏ n = 1 ∞ ( 1 − q n ) , q = e 2 π i z . {\displaystyle \eta (z)=q^{1/24}\prod _{n=1}^{\infty
Modular_form
Transcendental single-variable function
tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred
Clausen_function
Sporadic simple group
2q^{3}+10698752q^{4}+\cdots \end{aligned}}} and η(τ) is the Dedekind eta function. Wilson (1999) found the 30 conjugacy classes of maximal subgroups of
Baby_monster_group
Symbols for constants, special functions
used often in colloidal chemistry H represents: the Eta function of Ludwig Boltzmann's H-theorem ("Eta" theorem), in statistical mechanics Information theoretic
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Infinite series with alternating signs
functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function. The series' terms (1, −2, 3, −4, ...) do not approach
1_−_2_+_3_−_4_+_⋯
Basque separatist group (1960–2018)
ETA, an acronym for Euskadi Ta Askatasuna ('Basque Homeland and Liberty' or 'Basque Country and Freedom' in Basque), was an armed Basque nationalist and
ETA_(separatist_group)
Generalized function whose value is zero everywhere except at zero
\eta _{\varepsilon }*\eta _{\delta }=\eta _{\varepsilon +\delta }} for all ε, δ > 0. Convolution semigroups in L1 that approximate the delta function are
Dirac_delta_function
Sporadic simple group
(sequence A007251 in the OEIS) and η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta function. Norton and Wilson found the 14 conjugacy classes of maximal
Harada–Norton_group
Mathematical technique in thermal field theory
S_{\eta }={\frac {1}{\beta }}\sum _{i\omega }g(i\omega )={\frac {1}{2\pi i\beta }}\oint g(z)h_{\eta }(z)\,dz,} As in Fig. 1, the weighting function generates
Matsubara_summation
Concept in economics
η {\displaystyle \eta } is a constant that is positive for risk averse agents. Since additive constant terms in objective functions do not affect optimal
Isoelastic_utility
Family of probability distributions related to the normal distribution
\right)}=h(x)\,\exp \left[\eta (\theta )\cdot T(x)-A(\theta )\right]} where T(x), h(x), η(θ), and A(θ) are known functions. The function h(x) must be non-negative
Exponential_family
Mathematical function
Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
Ramanujan_theta_function
Indian mathematician (1887–1920)
|ℜ(θ)|<π and |ℑ(θ)|<π, where Γ(z) is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating
Srinivasa_Ramanujan
Natural number
are given by the Ramanujan τ-function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function: Δ(τ)=(2π)12η24(τ) This fact
12_(number)
signature defect of the boundary of a manifold as the eta invariant, the value as s = 0 of their eta function, and used this to show that Hirzebruch's signature
Signature_defect
Continuous probability distribution, named after Benjamin Gompertz
function of the Gompertz distribution is: f ( x ; η , b ) = b η exp ( η + b x − η e b x ) for x ≥ 0 , {\displaystyle f\left(x;\eta ,b\right)=b\eta
Gompertz_distribution
Symmetric holomorphic function
)=k^{2}(\tau )} . In terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} and theta functions, λ ( τ ) = ( 2 η ( τ 2 ) η 2 ( 2 τ ) η 3
Modular_lambda_function
Ability of an imaging system to resolve detail
\operatorname {sinc} (\xi ,\eta )} function corresponding to the active area. That last function serves as an overall envelope to the MTF function; so long as the
Optical_resolution
Function that maps matrices to matrices
{\begin{aligned}f(a+\eta b)&=f(a)+f'(a){\frac {\eta b}{1!}}+f''(a){\frac {(\eta b)^{2}}{2!}}+f'''(a){\frac {(\eta b)^{3}}{3!}}\\[.5em]&=a^{3}+3a^{2}(\eta b)+3a(\eta b)^{2}+(\eta
Analytic_function_of_a_matrix
Sporadic simple group
}+1956q^{4}+5135q^{5}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. It can be defined in terms of the generators a and b and relations a
Held_group
Sporadic simple group
5367q^{3}+371520q^{4}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes of maximal
Fischer_group_Fi23
American mathematician
which he gave a first solution to Landau's problem on the Dirichlet eta function), An introduction to transform theory, and The convolution transform
David_Widder
Function related to statistics and probability theory
{\boldsymbol {\eta }}} and the sufficient statistic T ( x ) {\displaystyle \mathbf {T} (x)} , minus the normalization factor (log-partition function) A (
Likelihood_function
Stellar system in the constellation Carina
Eta Carinae (η Carinae, abbreviated to η Car), formerly known as η Argus, is a stellar system containing at least two stars with a combined luminosity
Eta_Carinae
z ) {\displaystyle \eta (z)} denote the Dedekind eta function. Then for q = e 2 π i z {\displaystyle q=e^{2\pi iz}} , the function S ~ ( z ) := q − 1 /
Spt_function
Continued fraction closely related to the Rogers–Ramanujan identities
throughout this section since the q-expansion of the j-function (as well as the well-known Dedekind eta function) uses q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau
Rogers–Ramanujan continued fraction
Rogers–Ramanujan_continued_fraction
Special mathematical function
last identity was derived by Malmsten in 1842. Hurwitz zeta function Dirichlet eta function Polylogarithm Dirichlet Beta – Hurwitz zeta relation, Engineering
Dirichlet_beta_function
eigenvalue Dirichlet's ellipsoidal problem Dirichlet eta function (number theory) Dirichlet form Dirichlet function (topology) Dirichlet hyperbola method Dirichlet
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Infinite product for pi
{\displaystyle k\rightarrow \infty } . The Riemann zeta function and the Dirichlet eta function can be defined: ζ ( s ) = ∑ n = 1 ∞ 1 n s , ℜ ( s ) > 1
Wallis_product
German mathematician (1877–1938)
inequality Landau–Ramanujan constant Landau's problem on the Dirichlet eta function Landau kernel Endmund Landau (1895). "Zur relativen Wertbemessung der
Edmund_Landau
In physics, solution to Schrödinger equation
hypergeometric function, η = Z m c α / ( ℏ k ) {\displaystyle \eta =Zmc\alpha /(\hbar k)} and Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The two
Coulomb_wave_function
factorization theorem Weierstrass function (nowhere-differentiable function) Weierstrass sigma, zeta, and eta functions Weierstrass M-test Weierstrass Nullstellensatz
List of things named after Karl Weierstrass
List_of_things_named_after_Karl_Weierstrass
Special type of functions in mathematics
{\displaystyle S_{mn}(c,\eta )=(1-\eta ^{2})^{m/2}Y_{mn}(c,\eta )} , the function Y m n ( c , η ) {\displaystyle Y_{mn}(c,\eta )} satisfies ( 1 − η 2
Prolate spheroidal wave function
Prolate_spheroidal_wave_function
Type of generalization of periodic functions in Euclidean space
harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G {\displaystyle G} to the complex numbers (or
Automorphic_form
Array of nonnegative integers in combinatorics
{S}}_{2}}q^{|\pi |}=\prod _{\eta \in {\mathcal {B}}(r,r,t)/{\mathcal {S}}_{2}}{\frac {1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}}} In 1972 Edward A
Plane_partition
Second-order differential operator
{\displaystyle \eta _{00}=1} , η 11 = η 22 = η 33 = − 1 {\displaystyle \eta _{11}=\eta _{22}=\eta _{33}=-1} , η μ ν = 0 {\displaystyle \eta _{\mu \nu }=0}
D'Alembert_operator
Special mathematical function
to Dirichlet eta function and the Dirichlet beta function: Li s ( − 1 ) = − η ( s ) , {\displaystyle \operatorname {Li} _{s}(-1)=-\eta (s),} where η(s)
Polylogarithm
Class of statistical models
distributions, 2. A linear predictor η = X β {\displaystyle \eta =X\beta } , and 3. A link function g {\displaystyle g} such that E ( Y ∣ X ) = μ = g − 1
Generalized_linear_model
Generating function for quantum correlation functions
Grassmann currents η {\displaystyle \eta } and η ¯ {\displaystyle {\bar {\eta }}} so that the partition function is Z [ η ¯ , η ] = ∫ D ψ ¯ D ψ e i
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
German company
ETAS GmbH is a German company which designs tools for the development of embedded systems for the automotive industry and other sectors of the embedded
ETAS
Mathematical integral
{\displaystyle F_{j}(0)=\eta (j+1),} where η {\displaystyle \eta } is the Dirichlet eta function. Incomplete Fermi–Dirac integral Gamma function Polylogarithm Fermi–Dirac
Complete_Fermi–Dirac_integral
Generalizations of the Riemann zeta function
{H}}_{n}^{(c)}}{(n+1)^{b}}}=\zeta (a,b,{\bar {c}})} As a variant of the Dirichlet eta function we define ϕ ( s ) = 1 − 2 ( s − 1 ) 2 ( s − 1 ) ζ ( s ) {\displaystyle
Multiple_zeta_function
Probability distribution
{\displaystyle \eta } . It is more flexible than the Gumbel distribution. The hazard rate is a concave function of F ( x ; b , η ) {\displaystyle F(x;b,\eta )} which
Shifted_Gompertz_distribution
Algebraic curve in mathematics
)^{2}=(2\pi )^{12}\,\eta ^{24}(\tau )} is generally a transcendental number. In particular, the value of the Dedekind eta function η(2i) is η ( 2 i ) =
Elliptic_curve
Prime number with a certain relationship to an elliptic curve
{\displaystyle \eta (\tau )^{2}\eta (11\tau )^{2}} vanishes modulo p {\displaystyle p} , where η {\displaystyle \eta } is the Dedekind eta function. More generally
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
Concept in algebraic number theory
cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the
Heegner_number
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
{\displaystyle \operatorname {\Im } \tau >0} and η {\displaystyle \eta } is the eta function. The above result can be equivalently written as ∑ n = 1 ∞ ν (
Lemniscate_constant
26-dimensional string theory
_{1}<{\frac {1}{2}}\right\}} . η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta function. The integrand is of course invariant under the modular
Bosonic_string_theory
German mathematician (1831–1916)
Dedekind domain Dedekind eta function Dedekind-infinite set Dedekind number Dedekind psi function Dedekind sum Dedekind zeta function Ideal (ring theory) "Dedekind"
Richard_Dedekind
Computer programming function
transformations correspond to functions of the form eta :: F a -> G a, where a is a universally quantified type variable – eta knows nothing about the type
Map_(higher-order_function)
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Part of signal analysis and signal processing
distribution function if a kernel function Φ ( η , τ ) ≠ 1 {\displaystyle \Phi (\eta ,\tau )\neq 1} is chosen. A properly chosen kernel function can significantly
Bilinear time–frequency distribution
Bilinear_time–frequency_distribution
Modular form
the Dedekind eta function. The Fourier coefficients here are written τ ( n ) {\displaystyle \tau (n)} and called 'Ramanujan's tau function', with the normalization
Cusp_form
Probability distribution
{\displaystyle V_{p}(x,f)=\eta \cdot L(x,f)+(1-\eta )\cdot G(x,f)} with 0 < η < 1 {\displaystyle 0<\eta <1} . η {\displaystyle \eta } is a function of full width at
Voigt_profile
Growth curve model
W n {\displaystyle {\frac {dW}{dt}}=\eta W^{m}-\kappa W^{n}} with W {\textstyle W} the weight, η {\textstyle \eta } and κ {\textstyle \kappa } constants
Von_Bertalanffy_function
Two-dimensional laminar boundary layer that forms on a semi-infinite plate
stream function. The stream function is directly proportional to the normalized function, f ( η ) {\displaystyle f(\eta )} , which is only a function of the
Blasius_boundary_layer
relation Cyclotomic polynomials H. G. Dawson: Dawson function Richard Dedekind: Dedekind eta function Charles F. Dunkl: Dunkl operator, Jacobi–Dunkl operator
List of eponyms of special functions
List_of_eponyms_of_special_functions
Second-order partial differential equation describing motion of mechanical system
small and η {\displaystyle \eta } is a differentiable function satisfying η ( a ) = η ( b ) = 0 {\displaystyle \eta (a)=\eta (b)=0} . Then define Φ ( ε
Euler–Lagrange_equation
British-Lebanese mathematician (1929–2019)
Segal on twisted K-theory. One paper is a detailed study of the Dedekind eta function from the point of view of topology and the index theorem. Several of
Michael_Atiyah
Plane algebraic curve
McKay–Thompson series for the class 2B of the Monster, and η is the Dedekind eta function, then x = ( j 2 + 256 ) 3 j 2 2 , {\displaystyle x={\frac {(j_{2}+256)^{3}}{j_{2}^{2}}}
Classical_modular_curve
Theorem in number theory
Euler's function, which is closely related to the Dedekind eta function, and occurs in the study of modular forms. The modulus of the Euler function (see
Pentagonal_number_theorem
Summation method for some divergent series
summation to the zeta function (or rather, to the related Dirichlet eta function) yields (cf. Globally convergent series) 1 1 − 2 k + 1 ∑ i = 0 k 1 2
Euler_summation
Special mathematical function
{1}{n^{s}}}=\Phi (1,s,1)} The Dirichlet eta function: η ( s ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n s = Φ ( − 1 , s , 1 ) {\displaystyle \eta (s)=\sum _{n=1}^{\infty }{\frac
Lerch_transcendent
Mathematical-logic system
POW′′ := λb.λn.λf.n b f but that is just an eta-expanded version of POW we already have, above. The predecessor function, specified by two equations PRED (SUCC
Lambda_calculus
Infinite series summing alternating 1 and -1 terms
Dirichlet series for η defines a function on the whole complex plane — the Dirichlet eta function — and moreover, this function is analytic. For z with real
Grandi's_series
Canadian mathematician (1953–2020)
developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision
Peter_Borwein
Infinite series whose terms alternate in sign
{x}{2}}\right)}^{2m+\alpha }} where Γ(z) is the gamma function. If s is a complex number, the Dirichlet eta function is formed as an alternating series η ( s ) =
Alternating_series
Varying methods used to calculate pi
but this time is a quotient of a modular form, namely the Dedekind eta function, and where the argument involves τ = − 3502 {\displaystyle \tau ={\sqrt
Approximations_of_pi
Unsolved problem in mathematics
(z)=q\prod _{n>0}\left(1-q^{n}\right)^{24}=\eta (z)^{24},} where η ( z ) {\textstyle \eta (z)} is the Dedekind eta function. Δ ( z ) {\textstyle \Delta (z)} is
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Computational statistics technique
cumulant-generating function as ψ θ ( η ) = ψ ( θ + η ) − ψ ( θ ) = ( μ + θ σ 2 ) η + σ 2 η 2 2 {\textstyle \psi _{\theta }(\eta )=\psi (\theta +\eta )-\psi (\theta
Rejection_sampling
travel, tourism, insurance
ETA FUNCTION
ETA FUNCTION
ETA FUNCTION
ETA FUNCTION
ETA FUNCTION
ETA FUNCTION
ETA FUNCTION
ETA FUNCTION
ETA FUNCTION
travel, tourism, insurance