Searches , social queries for RECIPROCAL GAMMA-FUNCTION
Search references for RECIPROCAL GAMMA-FUNCTION. Phrases containing RECIPROCAL GAMMA-FUNCTION
See searches and references containing RECIPROCAL GAMMA-FUNCTION!
Search references for RECIPROCAL GAMMA-FUNCTION. Phrases containing RECIPROCAL GAMMA-FUNCTION
See searches and references containing RECIPROCAL GAMMA-FUNCTION!RECIPROCAL GAMMA-FUNCTION
Mathematical function
reciprocal gamma function is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the gamma function.
Reciprocal_gamma_function
Extension of the factorial function
Since the gamma function has no zeros, its reciprocal 1 / Γ {\displaystyle 1/\Gamma } is an entire function. In fact, the gamma function corresponds to
Gamma_function
Inverse of the gamma function
mathematics, the inverse gamma function Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gamma function. In other words, y = Γ
Inverse_gamma_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
Function that is holomorphic on the whole complex plane
sigma function. Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and
Entire_function
Mathematical constant
mathematical constant that represents the area between the graph of the reciprocal Gamma function, 1/Γ, and the positive x axis. That is, F = ∫ 0 ∞ 1 Γ ( x ) d
Fransén–Robinson_constant
Two-parameter family of continuous probability distributions
distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in
Inverse-gamma_distribution
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Mathematical function
In mathematics, the nu function is a generalization of the reciprocal gamma function of the Laplace transform. Formally, it can be defined as ν ( x )
Nu_function
Analytic function in mathematics
{d} x} is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ
Riemann_zeta_function
Special functions of several complex variables
{\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt[{3}]{{\sqrt[{4}]{2}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{216}}}}\end{aligned}}} If the reciprocal of the Gelfond
Theta_function
Types of special mathematical functions
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to certain integrals. Their respective
Incomplete_gamma_function
Complex-differentiable (mathematical) function
{\displaystyle \mathbb {C} \smallsetminus \{0\}} . (The reciprocal function, and any other rational function, is meromorphic on C {\displaystyle \mathbb {C}
Holomorphic_function
Meromorphic function
\mathbb {C} } defined as the (m + 1)th derivative of the logarithm of the gamma function: ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z )
Polygamma_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
Operation on formal power series
factorial function example given immediately below in this section. The last integral formula is compared to Hankel's loop integral for the reciprocal gamma function
Generating function transformation
Generating_function_transformation
Constants of the mathematical zeta function
/4)}}-{\frac {\Gamma '(1/2)}{\Gamma (1/2)}}=\log(2\pi )+{\frac {\pi }{2}}+2\log 2+\gamma \,.} The following sums can be derived from the generating function: ∑ k
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
the entire function defined by means of the reciprocal gamma function, then the Bessel–Clifford function is defined by the series C n ( z ) = ∑ k = 0
Bessel–Clifford_function
Image luminance mapping function
color use gamma 2.8. In most computer display systems, images are encoded with a gamma of about 0.45 and decoded with the reciprocal gamma of 2.2. A notable
Gamma_correction
Relations between flows and forces, or gradients, in thermodynamic systems
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium
Onsager_reciprocal_relations
Fundamental trigonometric functions
ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as: tan ( θ ) = sin ( θ ) cos
Sine_and_cosine
Analytic function that does not satisfy a polynomial equation
logarithm and inverse trigonometric functions. All special functions such as the gamma, error, bessel, and Riemann zeta functions are transcendental. Equations
Transcendental_function
Smooth nowhere-analytic function
\{0\}}\left(-{\frac {\Gamma (-\chi _{k})\zeta (1-\chi _{k})}{L}}\right)e^{2\pi ikt},} where Γ {\displaystyle \Gamma } is the gamma function and ζ {\displaystyle
Fabius_function
Probability theory
distribution of the reciprocal, Y = 1 / X. If the distribution of X is continuous with density function f(x) and cumulative distribution function F(x), then the
Inverse_distribution
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
Difference between logarithm and harmonic series
the reciprocal of the gamma function: 1 Γ ( z ) = z e γ z ∏ n = 1 ∞ ( e − z n ( 1 + z n ) ) . {\displaystyle {\frac {1}{\Gamma (z)}}=ze^{\gamma z}\prod
Euler's_constant
mathematics and number theory, the sum of reciprocals (or sum of inverses) is defined as the sum of reciprocals of some series of positive integers (counting
List_of_sums_of_reciprocals
W ) {\displaystyle T(W)} —the reciprocal of absolute risk aversion A ( W ) {\displaystyle A(W)} —is a linear function of wealth W: T ( W ) = 1 A ( W
Hyperbolic absolute risk aversion
Hyperbolic_absolute_risk_aversion
Number of integers coprime to and less than n
{\displaystyle \gamma } is Euler's constant and p 120569 # {\displaystyle p_{120569}\#} is the product of the first 120569 primes. Carmichael function (λ) Dedekind
Euler's_totient_function
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
Statistical function that defines the quantiles of a probability distribution
prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function. Consider a statistical application where a user
Quantile_function
Mathematical function
Reversing the order of the two letters of the function name results in the reciprocals of the three functions above: ns ( u ) = 1 sn ( u ) , nc ( u
Jacobi_elliptic_functions
Special mathematical function defined as sin(x)/x
}\left(1-{\frac {x^{2}}{n^{2}}}\right)} and is related to the gamma function Γ(x), as well as to Gauss' Pi function, through Euler's reflection formula: sin ( π x
Sinc_function
Fourier expansion of a reciprocal square root
{1}{2}})}}Q_{m-{\frac {1}{2}}}^{n}(z)e^{im\psi },} where Γ {\displaystyle \Gamma } is the gamma function. Heine (1881), p. 286. Cohl et al. (2000). Cohl (2003). Cohl
Heine's_identity
Product of numbers from 1 to n
factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Many other notable functions and
Factorial
Quantity in relativistic physics
definition, some authors define the reciprocal α = 1 γ = 1 − v 2 c 2 = 1 − β 2 ; {\displaystyle \alpha ={\frac {1}{\gamma }}={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\
Lorentz_factor
Discrete probability distribution
using the lgamma function in the C standard library (C99 version) or R, the gammaln function in MATLAB or SciPy, or the log_gamma function in Fortran 2008
Poisson_distribution
Notion from the theory of entire functions
)\right|} Another easily deducible indicator function is that of the reciprocal Gamma function. However, this function is of infinite type (and of order ρ =
Indicator function (complex analysis)
Indicator_function_(complex_analysis)
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
{z}{((m-{\frac {1}{2}})\pi )^{2}}}\right)} Another example is the reciprocal gamma function 1/Γ(z). It is the limit of polynomials as follows: 1 / Γ ( z )
Laguerre–Pólya_class
α + β + γ = 180 ∘ , {\displaystyle \alpha +\beta +\gamma =180^{\circ },} as long as the functions occurring in the formulae are well-defined (the latter
List of trigonometric identities
List_of_trigonometric_identities
Approximation for factorials
= Γ ( n + 1 ) , {\displaystyle n!=\Gamma (n+1),} where Γ denotes the gamma function. However, the gamma function, unlike the factorial, is more broadly
Stirling's_approximation
Arithmetic function
the Riemann zeta function evaluated at 2, which is π 2 6 {\displaystyle {\frac {\pi ^{2}}{6}}} . The summatory function of the reciprocal of the totient
Totient_summatory_function
Special mathematical function
(Vepstas 2008). Bose integral is result of multiplication between Gamma function and Zeta function. One can begin with equation for Bose integral, then use series
Polylogarithm
Mathematical constant
$3.14159 billion (π). Divergence of the sum of the reciprocals of the primes Prime zeta function "Google's strange bids for Nortel patents". Financial
Meissel–Mertens_constant
Fourier transform of the probability density function
Zeyang; Xiang, Min; Mandic, Danilo (2020). "Reciprocal Adversarial Learning via Characteristic Functions". Advances in Neural Information Processing Systems
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Partial differential equations
{x} )} . The free-space Green's function for the Laplace operator in three variables is given in terms of the reciprocal distance between two points and
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
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
Formal power series
special functions and enumerate partition functions. In particular, we recall that the partition function p(n) is generated by the reciprocal infinite
Generating_function
Type of function in mathematics
special functions are analytic on a suitable domain: hypergeometric functions on suitable domains Bessel functions on suitable domains The gamma function away
Analytic_function
Constants in the zeta function's Laurent series expansion
the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function: ζ ( 1 + s ) = 1 s + ∑ n = 0 ∞ (
Stieltjes_constants
Fundamental theorem in condensed matter physics
(k + K), where K is any reciprocal lattice vector (see figure at right). Therefore, wave vectors that differ by a reciprocal lattice vector are equivalent
Bloch's_theorem
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
Family of continuous probability distributions
{\gamma (k,\lambda x)}{\Gamma (k)}}={\frac {\gamma (k,\lambda x)}{(k-1)!}},} where γ {\displaystyle \gamma } is the lower incomplete gamma function and
Erlang_distribution
Probability distribution
{(k+r-1)(k+r-2)\dotsm (r)}{k!}}={\frac {\Gamma (k+r)}{k!\ \Gamma (r)}}=\left(\!\!{r \choose k}\!\!\right).} Note that Γ(r) is the Gamma function, and ( ( r k ) ) {\displaystyle
Negative binomial distribution
Negative_binomial_distribution
Probability distribution
Further, Γ {\displaystyle \Gamma } is the gamma function. The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape
Inverse-chi-squared distribution
Inverse-chi-squared_distribution
Sum of inverse squares of natural numbers
the problem. The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite
Basel_problem
Symbols for constants, special functions
optical mode in a waveguide the gamma function, a generalization of the factorial the upper incomplete gamma function the modular group, the group of
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
SI unit of frequency
units is 1/s or s−1, meaning that one hertz is one per second or the reciprocal of one second. It is used only in the case of periodic events. It is named
Hertz
Primitive cell in the reciprocal space lattice of crystals
primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is
Brillouin_zone
Concept in complex analysis
holomorphic functions of a complex variable. For example, consider the reciprocal function, g(z) = 1/z which is holomorphic on the punctured plane C\{0}. A
Antiderivative (complex analysis)
Antiderivative_(complex_analysis)
Rules for computing derivatives of functions
the reciprocal rule. The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f {\textstyle
Differentiation_rules
Notation system for crystal lattice planes
based on the fact that a reciprocal lattice vector g (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector
Miller_index
Topic in probability theory and statistics
parameter p. A gamma distribution with shape parameter α = 1 and rate parameter β is an exponential distribution with rate parameter β. A gamma distribution
Relationships among probability distributions
Relationships_among_probability_distributions
Probability distribution
as a kernel Gaussian function Normally distributed and uncorrelated does not imply independent Ratio normal distribution Reciprocal normal distribution
Normal_distribution
Representation of a type of random process
{\begin{bmatrix}\gamma _{1}\\\gamma _{2}\\\gamma _{3}\\\vdots \\\gamma _{p}\\\end{bmatrix}}={\begin{bmatrix}\gamma _{0}&\gamma _{-1}&\gamma _{-2}&\cdots \\\gamma _{1}&\gamma
Autoregressive_model
Sum of the inverses of the positive cubes
spanning trees and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient, which appear occasionally
Apéry's_constant
Smooth function in statistics
for Normal, Bernoulli, Poisson, and Gamma. In addition, we describe the applications and use of variance functions in maximum likelihood estimation and
Variance_function
Mathematical functions
{2}}\pi ^{\frac {3}{2}}}{2\left(\Gamma \left({\frac {3}{4}}\right)\right)^{2}}}=2.62205\ldots } The lemniscate functions satisfy the basic relation cl
Lemniscate_elliptic_functions
Rational numbers in a reciprocal logarithm
Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the
Gregory_coefficients
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: H n = 1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k
Harmonic_number
Family of probability distributions related to the normal distribution
first need to expand the part of the log-partition function that involves the multivariate gamma function: log Γ p ( a ) = log ( π p ( p − 1 ) 4 ∏ j =
Exponential_family
and Γ {\displaystyle \Gamma } the corresponding reciprocal lattice. The theorem of Petersen and Middleton states that a function f ( ⋅ ) {\displaystyle
Multidimensional_sampling
{1}{x}}}}{2\gamma x}}\phi \left({\frac {{\sqrt {x}}-{\sqrt {\frac {1}{x}}}}{\gamma }}\right)\quad x>0;\gamma >0} Since the general form of probability functions can
Birnbaum–Saunders distribution
Birnbaum–Saunders_distribution
Radius of the circle which best approximates a curve at a given point
In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best
Radius_of_curvature
Mathematical function
{\displaystyle \Gamma (z)} is the gamma function. The final expression is defined for all complex numbers except the negative even integers, and its reciprocal is
Double_factorial
Value approached by a mathematical object
trajectory to be a function γ : R → X {\displaystyle \gamma :\mathbb {R} \rightarrow X} , the point γ ( t ) {\displaystyle \gamma (t)} is thought of as
Limit_(mathematics)
Mathematical measure of how much a curve or surface deviates from flatness
is parametrized by arc length is a vector-valued function that is denoted by the Greek letter gamma with an overbar, –γ, that describes the position of
Curvature
Concept in combinatorics (part of mathematics)
The reciprocal of the function ( q ) ∞ := ( q ; q ) ∞ {\displaystyle (q)_{\infty }:=(q;q)_{\infty }} similarly arises as the generating function for the
Q-Pochhammer_symbol
cosine distribution on [ μ − s , μ + s {\displaystyle \mu -s,\mu +s} ] The reciprocal distribution The triangular distribution on [a, b], a special case of
List of probability distributions
List_of_probability_distributions
Values which describe behavior of a linear electric circuit
1 − S 22 Γ L | {\displaystyle |\Gamma _{\mathrm {in} }|=\left|S_{11}+{\frac {S_{12}S_{21}\Gamma _{L}}{1-S_{22}\Gamma _{L}}}\right|} and | Γ o u t | =
Scattering_parameters
Characteristic of any structure that is periodic across a position in space
repeat per unit of distance. The SI unit of spatial frequency is the reciprocal metre (m−1), although cycles per meter (c/m) is also common. In image-processing
Spatial_frequency
Continuous probability distribution
\ \mu _{k}=\Gamma \left(1-{\frac {k}{\alpha }}\right)\ } where Γ ( z ) {\displaystyle \ \Gamma \left(z\right)\ } is the Gamma function. In particular:
Fréchet_distribution
Measured time difference as explained by relativity theory
v(t)={\frac {gt+v_{0}\gamma _{0}}{\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}}.} The proper time as a function of coordinate time is
Time_dilation
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
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
Equation relating transport coefficients to correlation functions
transport coefficient γ {\displaystyle \gamma } in terms of the integral of the equilibrium time correlation function of the time derivative of a corresponding
Green–Kubo_relations
Special function defined by an integral
derivative of the circle function is the negative product of the identical mapping function and the reciprocal of the circle function: d d ε 1 − ε 2 = − ε
Elliptic_integral
Class of probability distributions
conjugate prior distributions of NEF-QVF distributions are the normal, gamma, reciprocal gamma, beta, F-, and t- distributions. Again, these conjugate priors
Natural_exponential_family
Measure used in radio engineering and telecommunications
{\displaystyle \Gamma } can be defined as: Γ = V r V f . {\displaystyle \Gamma ={\frac {V_{r}}{V_{f}}}.} or Γ = Z L − Z 0 Z L + Z 0 {\displaystyle \Gamma ={Z_{L}-Z_{0}
Standing_wave_ratio
Motion of charged particles
{\displaystyle p=\gamma mv} : r c = p ⊥ | q | B = γ m v ⊥ | q | B {\displaystyle r_{\rm {c}}={\frac {p_{\perp }}{|q|B}}={\frac {\gamma mv_{\perp }}{|q|B}}}
Cyclotron_motion
Inverse of a finite difference/the antidifference for various functions
_{x}\Gamma (x)=(-1)^{x+1}\Gamma (x){\frac {\Gamma (1-x,-1)}{e}}+C} where Γ ( s , x ) {\displaystyle \Gamma (s,x)} is the incomplete gamma function and
List_of_indefinite_sums
Decomposition of periodic functions
{\displaystyle {\tfrac {n}{P}}} in the reciprocal units of x {\displaystyle x} . These series can represent functions that are just a sum of one or more frequencies
Fourier_series
Concept in physics
wavevector in the reciprocal-space (Brillouin zone), and u n k ( r ) {\displaystyle u_{n\mathbf {k} }(\mathbf {r} )} is a periodic function of r {\displaystyle
Berry connection and curvature
Berry_connection_and_curvature
Mathematical function, inverse of an exponential function
rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, ln(t)
Logarithm
Thermodynamic quantity
{\begin{aligned}&C_{P}={\frac {\gamma R}{\gamma -1}},&&C_{V}={\frac {R}{\gamma -1}}\\&\gamma ={\frac {C_{P}}{C_{P}-R}},&&\gamma =1+{\frac {R}{C_{V}}}\end{aligned}}}
Heat_capacity_ratio
Family of probability distributions
)].} Here the minus exponent in τ−1(μ) denotes an inverse function rather than a reciprocal. The mean and variance of an additive random variable is then
Tweedie_distribution
Principle in kinetic systems
_{r}w_{r}^{\rm {eq}}\gamma _{ri}\gamma _{rj}} These symmetry relations, L i j = L j i {\displaystyle L_{ij}=L_{ji}} , are exactly the Onsager reciprocal relations
Detailed_balance
Number of subsets of a given size
generalized to two real or complex valued arguments using the gamma function or beta function via ( x y ) = Γ ( x + 1 ) Γ ( y + 1 ) Γ ( x − y + 1 ) = 1 (
Binomial_coefficient
Electronic circuit in which a signal entering any port exits at the next port
In electrical engineering, a circulator is a passive, non-reciprocal three- or four-port device that only allows a microwave or radio-frequency (RF) signal
Circulator
Function in quantum field theory showing probability amplitudes of moving particles
{1}{2}}(\gamma _{\mu }p^{\mu }\gamma _{\nu }p^{\nu }+\gamma _{\nu }p^{\nu }\gamma _{\mu }p^{\mu })\\[6pt]&={\tfrac {1}{2}}(\gamma _{\mu }\gamma _{\nu }+\gamma
Propagator
travel, tourism, insurance
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
RECIPROCAL GAMMA-FUNCTION
travel, tourism, insurance