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Mathematic function
mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely
Elliptic_gamma_function
Extension of the factorial function
Cahen–Mellin integral Elliptic gamma function Lemniscate constant Pseudogamma function Hadamard's gamma function Inverse gamma function Lanczos approximation
Gamma_function
Generalization of the Euler gamma function and the Barnes G-function
related to the q-gamma function, and triple gamma functions Γ 3 {\displaystyle \Gamma _{3}} are related to the elliptic gamma function. For ℜ a i > 0 {\displaystyle
Multiple_gamma_function
Analytic function on the upper half-plane with a certain behavior under the modular group
{\displaystyle \gamma } , see e.g. "DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions". dlmf.nist.gov
Modular_form
Class of periodic mathematical functions
analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because
Elliptic_function
Mathematical function
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as
Jacobi_elliptic_functions
Special functions of several complex variables
properties of elliptic curves?" and others, including abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions in two dimensions
Theta_function
function, Polygamma function Incomplete beta function Incomplete gamma function K-function Multivariate gamma function: A generalization of the Gamma
List of mathematical functions
List_of_mathematical_functions
Special function defined by an integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied
Elliptic_integral
hypergeometric series Elliptic gamma function Hahn–Exton q-Bessel function Jackson q-Bessel function q-exponential q-gamma function q-theta function Lists of mathematics
List_of_q-analogs
Mathematical functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied
Lemniscate_elliptic_functions
Mathematical function
forms. In particular the modular discriminant of the Weierstrass elliptic function with ω 2 = τ ω 1 {\displaystyle \omega _{2}=\tau \omega _{1}} can
Dedekind_eta_function
Mathematical constants
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer, half-integer, and
Particular values of the gamma function
Particular_values_of_the_gamma_function
Algebraic curve in mathematics
mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined
Elliptic_curve
Function defined by a hypergeometric series
non-negative integer, one has 2F1(z) → ∞. Dividing by the value Γ(c) of the gamma function, we have the limit: lim c → − m 2 F 1 ( a , b ; c ; z ) Γ ( c ) = (
Hypergeometric_function
Meromorphic function on the complex plane
{s+\kappa _{j}}{2}}\right)} where Γ {\displaystyle \textstyle \Gamma } denotes the gamma function, π {\displaystyle \textstyle \pi } denotes the automorphic
L-function
Mathematical equation
It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the period ratio τ, the
Picard–Fuchs_equation
Mathematical concept
{\displaystyle \Gamma } , as listed here.) Knowing the group structure of the singular fibers is useful for computing the Mordell-Weil group of an elliptic fibration
Elliptic_surface
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map
Dixon_elliptic_functions
Modular function in mathematics
the elliptic curve y 2 = 4 x 3 − g 2 ( τ ) x − g 3 ( τ ) {\displaystyle y^{2}=4x^{3}-g_{2}(\tau )x-g_{3}(\tau )} (see Weierstrass elliptic functions). Note
J-invariant
Concept in combinatorics (part of mathematics)
hypergeometric series Elliptic gamma function Jacobi theta function Lambert series Pentagonal number theorem q-derivative q-theta function q-Vandermonde identity
Q-Pochhammer_symbol
Analytic function that does not satisfy a polynomial equation
hyperbolic functions, and the inverses of all of these. Less familiar are the special functions of analysis, such as the gamma, elliptic, and zeta functions, all
Transcendental_function
Kind of complex manifold
Poincaré bundle Complex Lie group Automorphic function Intermediate Jacobian Elliptic gamma function Mumford, David (2008). Abelian varieties. C. P.
Complex_torus
Function for Heun's differential equation
In mathematics, the local Heun function H ℓ ( a , q ; α , β , γ , δ ; z ) {\displaystyle H\ell (a,q;\alpha ,\beta ,\gamma ,\delta ;z)} is the solution of
Heun_function
Symmetric holomorphic function
^{*}(x))} (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any x ∈ Q + {\displaystyle
Modular_lambda_function
Type of mathematical function
most special functions are not elementary. Non-elementary functions include: the gamma function non-elementary Liouvillian functions, including the
Elementary_function
Conformal mappings in complex analysis
(1-a')\Gamma (b)\Gamma (c')}{\Gamma (1-a)\Gamma (b')\Gamma (c)}},\end{aligned}}} where Γ ( x ) {\textstyle \Gamma (x)} is the gamma function. Near each
Schwarz_triangle_function
Type of differential operator
smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations
Elliptic_operator
disastrous consequences for applications of this function. The elliptic-curve version of this function is also of interest. In particular, it may help
Naor–Reingold pseudorandom function
Naor–Reingold_pseudorandom_function
Fundamental trigonometric functions
elliptic functions Euler's formula Generalized trigonometry Hyperbolic function Lemniscate elliptic functions Law of sines List of periodic functions
Sine_and_cosine
Mathematical function
particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In
Ramanujan_theta_function
Arithmetic function related to the divisors of an integer
series of the Eisenstein series and the invariants of the Weierstrass elliptic functions. For k > 0 {\displaystyle k>0} , there is an explicit series representation
Divisor_function
Swiss physicist and mathematician
mechanics and resulting special functions (such as the elliptic gamma function, elliptic quantum groups, and elliptic Macdonald polynomials). With Alberto
Giovanni_Felder
Analyzes the topology of a manifold by studying differentiable functions on that manifold
function on M {\displaystyle M} and p {\displaystyle p} is a non-degenerate critical point of f {\displaystyle f} of index γ , {\displaystyle \gamma
Morse_theory
Elliptic differential operators in geometry mathematics
In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Function in q-analog theory
{\displaystyle q} -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was
Q-gamma_function
Special function occurring in problems possessing elliptic symmetry
equation (PDE) boundary value problems possessing elliptic symmetry. In some usages, Mathieu function refers to solutions of the Mathieu differential equation
Mathieu_function
Differential operator in mathematics
an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The Laplace–Beltrami operator, when applied to a function, is
Laplace_operator
Family of power series in mathematics
(a)_{n}=a(a+1)(a+2)\cdots (a+n-1)={\frac {\Gamma (a+n)}{\Gamma (a)}}} where Γ {\displaystyle \Gamma } represents the gamma function. The series can then be written
Generalized hypergeometric function
Generalized_hypergeometric_function
Special function of two variables
analogue of a classical elliptic modular function. Note that E ( z , s ) {\displaystyle E(z,s)} is not a square-integrable function of z {\displaystyle z}
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
Flash of gamma rays from a distant galaxy
In gamma-ray astronomy, gamma-ray bursts (GRBs) are extremely energetic events occurring in distant galaxies that represent the brightest and most powerful
Gamma-ray_burst
Rational function of the form (az + b)/(cz + d)
{\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\begin{pmatrix}\gamma _{1}-k\gamma _{2}&(k-1)\gamma _{1}\gamma _{2}\\1-k&k\gamma _{1}-\gamma _{2}\end{pmatrix}}}
Möbius_transformation
Green's function for Laplacian
defined by convolution with a function having a mathematical singularity at the origin, the Newtonian kernel Γ {\displaystyle \Gamma } which is the fundamental
Newtonian_potential
In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g2 = 0
Equianharmonic
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
the lemniscate elliptic functions and is approximately equal to 2.62205755. It also appears in evaluation of the gamma and beta function at certain rational
Lemniscate_constant
Q-analog of hypergeometric series
generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If
Basic_hypergeometric_series
Major type of automorphic form in mathematics
automorphic form that generalize conventional elliptic modular forms, which are closely related to elliptic curves. The complex manifolds constructed in
Siegel_modular_form
Summability method in physics
to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization
Zeta_function_regularization
Sequence of numbers ((2n) choose (n))
{\displaystyle \Gamma (x)} is the gamma function and B ( x , y ) {\displaystyle \mathrm {B} (x,y)} is the beta function. The powers of two that divide the
Central_binomial_coefficient
Mathematical idealization of the trace left by a moving point
{\displaystyle \gamma :[a,b]\to \mathbb {R} ^{n}} is an injective and continuously differentiable function, then the length of γ {\displaystyle \gamma } is defined
Curve
Exponential function Beta function Gamma function Riemann zeta function Riemann hypothesis Generalized Riemann hypothesis Elliptic function Half-period
List of complex analysis topics
List_of_complex_analysis_topics
Plane algebraic curve
the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals
Lemniscate_of_Bernoulli
Type of generalization of periodic functions in Euclidean space
< G {\displaystyle \Gamma <G} of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to
Automorphic_form
Mathematical functions that quantify complexity
Swinnerton-Dyer conjecture Elliptic Lehmer conjecture Heath-Brown–Moroz constant Height of a formal group law Height zeta function Raynaud's isogeny theorem
Height_function
Algebraic stack in mathematics
In mathematics, the moduli stack of elliptic curves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M e l l {\displaystyle {\mathcal {M}}_{\mathrm
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Orientation-preserving mapping class group of the torus
in GL(2, Z). It is for this reason that doubly periodic functions, such as elliptic functions, possess a modular group symmetry. The action of the modular
Modular_group
Probability distribution
is the number of degrees of freedom, and Γ {\displaystyle \Gamma } is the gamma function. This may also be written as f ( t ) = 1 ν B ( 1 2 , ν 2 ) (
Student's_t-distribution
Family of distributions that generalize the multivariate normal distribution
multivariate-statistical procedures. Elliptical distributions are defined in terms of the characteristic function of probability theory. A random vector
Elliptical_distribution
Mathematical functions having established names and notations
nineteenth century. The high point of special function theory in 1800–1900 was the theory of elliptic functions; treatises that were essentially complete
Special_functions
Number, approximately 3.14
with the identity Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!} . When the gamma function is evaluated at half-integers, the result is naturally
Pi
Function studied by Ramanujan
In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z}
Ramanujan_tau_function
Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals
Maximal_function
Riemann zeta function. Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. ψ n ( z ) {\displaystyle \psi _{n}(z)} is a polygamma function. Li s (
List_of_mathematical_series
Second-order partial differential equation
Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case
Laplace's_equation
Special functions in mathematics
differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second-order ordinary differential equations
Painlevé_transcendents
Algebraic variety
"best models" can be very different from those taken directly from elliptic function theory. Hecke operators may be studied geometrically, as correspondences
Modular_curve
Mathematical function
dy=2\pi A\sigma _{X}\sigma _{Y}.} In general, a two-dimensional elliptical Gaussian function is expressed as f ( x , y ) = A exp ( − ( a ( x − x 0 ) 2 +
Gaussian_function
Problem of solving a partial differential equation subject to prescribed boundary values
Green's function in two dimensions: G ( z , x ) = − 1 2 π log | z − x | + γ ( z , x ) , {\displaystyle G(z,x)=-{\frac {1}{2\pi }}\log |z-x|+\gamma (z,x)
Dirichlet_problem
Type of signal processing filter
in the passband than Chebyshev Type I/Type II and elliptic filters can achieve. A transfer function of a third-order low-pass Butterworth filter design
Butterworth_filter
Function related to statistics and probability theory
derivatives of the sufficient statistic T and the log-partition function A. The gamma distribution is an exponential family with two parameters, α {\textstyle
Likelihood_function
Mathematical algorithm
sufficiently regular boundary Γ {\displaystyle \Gamma } , let h be a function on Γ {\displaystyle \Gamma } , and let x {\displaystyle x} be a point inside
Walk-on-spheres_method
Symbols for constants, special functions
relational algebra the Pi function, i.e. the Gamma function when offset to coincide with the factorial the complete elliptic integral of the third kind
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Probability distribution
-1}\end{aligned}}} where Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The beta function, B {\displaystyle \mathrm {B} } , is a normalization
Beta_distribution
Conjecture on zeros of the zeta function
(n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}} is true for all n ≥ 120569#, where φ(n) is Euler's totient function and
Riemann_hypothesis
Function in quantum field theory showing probability amplitudes of moving particles
often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function). In non-relativistic quantum
Propagator
Mathematical theorem about the real analytic Eisenstein series
_{n\geq 1}(1-q^{n}p)(1-q^{n}/p).} Herglotz–Zagier function Serge Lang, Elliptic functions, ISBN 0-387-96508-4 C. L. Siegel, Lectures on advanced
Kronecker_limit_formula
Problem in celestial mechanics
hyperbolic and elliptic cases of the Lambert Problem. THORNE, JAMES (1990-08-17). "Series reversion/inversion of Lambert's time function". Astrodynamics
Lambert's_problem
Verification of special functions: Gamma function Elliptic functions Hypergeometric functions Hurwitz zeta function Bessel function Matrix function Verification
Validated_numerics
Mathematical manifold theory
( E N ) → 0 {\displaystyle 0\to \Gamma (E_{0})\to \Gamma (E_{1})\to \cdots \to \Gamma (E_{N})\to 0} is an elliptic complex. Introduce the direct sums:
Hodge_theory
German mathematician (born 1958)
involving the Gamma function: ζ ∞ ( s ) := 2 − 1 / 2 π − s / 2 Γ ( s / 2 ) . {\displaystyle \zeta _{\infty }(s):=2^{-1/2}\pi ^{-s/2}\Gamma (s/2).} More
Christopher_Deninger
{\displaystyle {\textrm {Decode}}_{\gamma }(\cdot )} which extracts the γ {\displaystyle \gamma } value from an encoding. A hash function H n ( ⋅ ) {\displaystyle
Implicit_certificate
Integrals not expressible in closed-form from elementary functions
{1-x^{4}}}} (elliptic integral) 1 ln x {\displaystyle {\frac {1}{\ln x}}} (logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}} (error function, Gaussian
Nonelementary_integral
Connects non-singular algebraic curves with compact Riemann surfaces
elliptischen Functionen" [On the eleventh order transformation of elliptic functions]. Mathematische Annalen (in German). 15 (3–4): 533–555. doi:10.1007/BF02086276
Belyi's_theorem
Probability of survival beyond any specified time
certain time. The survival function is also known as the survivor function or reliability function. The term reliability function is common in engineering
Survival_function
Approximation method
equations, or solving elliptic partial differential equations, a rank proportional to log ( 1 / ϵ ) γ {\displaystyle \log(1/\epsilon )^{\gamma }} with a small
Hierarchical_matrix
\Gamma (z)} is the Gamma function) ∫ 0 1 ( ln 1 x ) p d x = Γ ( p + 1 ) {\displaystyle \int _{0}^{1}\left(\ln {\frac {1}{x}}\right)^{p}\,dx=\Gamma (p+1)}
Lists_of_integrals
Well defined hypergeometric series discovered by Giuseppe Lauricella
{E} (k)} are the complete elliptic integrals of the first and second kind, respectively. In analogy with Appell's function F1, Lauricella's FD can be
Lauricella hypergeometric series
Lauricella_hypergeometric_series
Graph drawing used to study Riemann surfaces
pairs is to replace a Belyi function β {\displaystyle \beta } by the pure Belyi function γ = 4 β ( 1 − β ) {\displaystyle \gamma =4\beta (1-\beta )} . One
Dessin_d'enfant
On eigenvalues of random matrices
(j;x)=\int _{x}^{\infty }t^{j-1}e^{-t}dt} denotes the upper incomplete gamma function. It has the following asymptotics K ∞ b ( w , z ) := lim N → ∞ K N (
Circular_law
Mathematical theorem
\int _{G^{\gamma }\setminus G}\phi (x^{-1}\gamma x)\,dx} are orbital integrals. Define the following operator on compactly supported functions on Γ ∖ G
Selberg_trace_formula
Lemma in numerical analysis of differential equations
tool for proving error estimates for the finite element method applied to elliptic partial differential equations. Let V {\displaystyle V} be a real Hilbert
Céa's_lemma
Number, approximately 0.916
Malmsten's integrals. If K(k) is the complete elliptic integral of the first kind, as a function of the elliptic modulus k, then[citation needed] G = 1 2 ∫
Catalan's_constant
Type of partial differential equation
for both an unknown function u {\displaystyle u} and an unknown domain Ω {\displaystyle \Omega } . The segment Γ {\displaystyle \Gamma } of the boundary
Free_boundary_problem
Australian mathematician and computer scientist
Euler–Mascheroni constant γ {\displaystyle \gamma } using Bessel functions, and showed that γ {\displaystyle \gamma } can not have a simple rational form p/q
Richard_P._Brent
Distance along a curve
closed form solution for the arc length of an elliptic and hyperbolic arc led to the development of the elliptic integrals. In most cases, including even simple
Arc_length
Mathematical conjecture about zeros of L-functions
proven occur in the algebraic function field case (not the number field case). Global L-functions can be associated to elliptic curves, number fields (in
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Modular form
right side of the Weierstrass equation of an elliptic curve; and the 24-th power of the Dedekind eta function. The Fourier coefficients here are written
Cusp_form
Numbers expressible as integrals of algebraic functions
a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes
Period_(number_theory)
z , τ ) {\displaystyle \vartheta (z,\tau )} is not an ordinary function on the elliptic curve E τ = C / ( Z + τ Z ) , {\displaystyle E_{\tau }=\mathbb
Theta_representation
1964 mathematical reference work edited by M. Abramowitz and I. Stegun
author of the Gamma function section and other sections of the book Louis Melville Milne-Thomson, author of the book chapters on elliptic integrals and
Abramowitz_and_Stegun
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ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
ELLIPTIC GAMMA-FUNCTION
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