Reciprocal Gamma function

Reciprocal Gamma function

In mathematics, the reciprocal Gamma function is the function

:f(z) = frac{1}{Gamma(z)},

where Gamma(z) denotes the Gamma function. Since the Gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. 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.

Taylor series

Taylor series expansion around 0 gives

:frac{1}{Gamma(z)} = z + gamma z^2 + left(frac{gamma^2}{2} - frac{pi^2}{12} ight)z^3 + ldots

where gamma is the Euler-Mascheroni constant. For "k" > 2, the coefficient "ak" for the "zk" term can be computed recursively as

:a_k = k a_1 a_k - a_2 a_{k-1} + sum_{j=2}^{k-1} (-1)^j , zeta(j) , a_{k-j}

where ζ("s") is the Riemann zeta function.

Contour integral representation

An integral representation due to Hermann Hankel is

:frac{1}{Gamma(z)} = frac{i}{2pi} oint_C (-t)^{-z} e^{-t} dt,

where "C" is a path encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis. According to Schmelzer & Trefethen, numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the Gamma function.

Integral along the real axis

Integration of the reciprocal Gamma function along the positive real axis gives the value

:int_{0}^infty frac{1}{Gamma(x)} dx approx 2.80777024,

which is known as the Fransén-Robinson constant.

ee also

* Inverse-gamma distribution

References

* Thomas Schmelzer & Lloyd N. Trefethen, [http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/gamma.pdf Computing the Gamma function using contour integrals and rational approximations]
* Mette Lund, [http://www.nbi.dk/~polesen/borel/node14.html An integral for the reciprocal Gamma function]
* Milton Abramowitz & Irene A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables"
* Eric W. Weisstein, " [http://mathworld.wolfram.com/GammaFunction.html Gamma Function] ", MathWorld


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Gamma function — For the gamma function of ordinals, see Veblen function. The gamma function along part of the real axis In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its… …   Wikipedia

  • Particular values of the Gamma function — The Gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half integer arguments, but no simple expressions are known for the values at rational points in… …   Wikipedia

  • Gamma correction — Gamma correction, gamma nonlinearity, gamma encoding, or often simply gamma, is the name of a nonlinear operation used to code and decode luminance or tristimulus values in video or still image systems. [cite book | title = Digital Video and HDTV …   Wikipedia

  • Función gamma inversa — Gráfica de 1/Γ(x) a lo largo del eje real. Función gamma invers …   Wikipedia Español

  • Entire function — In complex analysis, an entire function, also called an integral function, is a complex valued function that is holomorphic over the whole complex plane. Typical examples of entire functions are the polynomials and the exponential function, and… …   Wikipedia

  • Riemann zeta function — ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): dark colors denote values close to zero and hue encodes the value s argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the… …   Wikipedia

  • Inverse-gamma distribution — Probability distribution name =Inverse gamma type =density pdf cdf parameters =alpha>0 shape (real) eta>0 scale (real) support =xin(0;infty)! pdf =frac{eta^alpha}{Gamma(alpha)} x^{ alpha 1} exp left(frac{ eta}{x} ight) cdf… …   Wikipedia

  • Bessel-Clifford function — In mathematical analysis, the Bessel Clifford function is an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. If :pi(x) = frac{1}{Pi(x)} = frac{1}{Gamma(x+1)}is… …   Wikipedia

  • Green's function for the three-variable Laplace equation — The free space Green s function for the three variable Laplace equation is given in terms of the reciprocal distance between two points. That is to say the solution of the equation : abla^2 G(mathbf{x},mathbf{x }) = delta(mathbf{x} mathbf{x }) is …   Wikipedia

  • List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”