In mathematics, the reciprocal gamma function is the function
where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that log log |1/Γ(z)| grows no faster than log |z|), but of infinite type (meaning that log |1/Γ(z)| grows faster than any multiple of |z|, since its growth is approximately proportional to |z| log |z| in the left-half plane).
The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function.
Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.
and 24 Related for: Reciprocal gamma function information
[clarification needed] The gammafunction has no zeros, so the reciprocalgammafunction 1/Γ(z) is an entire function. In fact, the gammafunction corresponds to...
The gammafunction is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer...
theta function, and the reciprocalGammafunction. The exponential function and the error function are special cases of the Mittag-Leffler function. According...
mathematics, the inverse gammafunction Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gammafunction. In other words, y = Γ...
In mathematics, the upper and lower incomplete gammafunctions are types of special functions which arise as solutions to various mathematical problems...
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gammafunction and to binomial...
{d} x} is the gammafunction. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ >...
logarithm log z is. The reciprocalfunction 1 / z is holomorphic on C ∖ {0}. (The reciprocalfunction, and any other rational function, is meromorphic on C...
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...
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium...
\mathbb {C} } defined as the (m + 1)th derivative of the logarithm of the gammafunction: ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z )...
color use gamma 2.8. In most computer display systems, images are encoded with a gamma of about 0.45 and decoded with the reciprocalgamma of 2.2. A notable...
{\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...
exponential function; others involve series or differential equations. From any of these definitions it can be shown that e−x is the reciprocal of ex. For...
_{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)} and is related to the gammafunction Γ(x) through Euler's reflection formula: sin ( π x ) π x = 1 Γ ( 1...
factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gammafunction. Many other notable functions and...
elementary functions. The indefinite integral of many algebraic functions is transcendental. For example, the logarithm function arose from the reciprocal 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...
/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...
the functional equation for the Gammafunction, Γ ( s ) Γ ( 1 − s ) = π sin ( π s ) , {\displaystyle \Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},}...
factorial function example given immediately below in this section. The last integral formula is compared to Hankel's loop integral for the reciprocalgamma function...
prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function. Consider a statistical application where a user...