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[[Image:Rgamma plot real.png|thumb|250px|Plot of 1/Γ(x) along the real axis]] | |||
[[Image:Complex Reciprocal Gamma.jpg|right|thumb|250px|Reciprocal gamma function 1/Γ(z) in the [[complex plane]]. The color of a point z encodes the value of 1/Γ(z). Strong colors denote values close to zero and hue encodes the value's [[complex number|argument]].]] | |||
In [[mathematics]], the '''reciprocal gamma function''' is the [[special function|function]] | |||
:<math>f(z) = \frac{1}{\Gamma(z)},</math> | |||
where Γ(''z'') denotes the [[gamma function]]. Since the gamma function is [[meromorphic function|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 <math>\log(\log|1/\Gamma(z)|)</math> grows no faster than <math>\log|z|</math>), but of infinite type (meaning that <math>\log|1/\Gamma(z)|</math> grows faster than any multiple of |z|, since its growth is approximately proportional to <math>|z|\log|z|</math> in the left-hand plane). | |||
The reciprocal is sometimes used as a starting point for [[numerical analysis|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 | |||
:<math>\frac{1}{\Gamma(z)} = z + \gamma z^2 + \left(\frac{\gamma^2}{2} - \frac{\pi^2}{12}\right)z^3 + \cdots</math> | |||
where γ is the [[Euler–Mascheroni constant]]. For ''k'' > 2, the coefficient ''a<sub>k</sub>'' for the ''z<sup>k</sup>'' term can be computed recursively as | |||
:<math>a_k = \frac{{- a_2 a_{k-1} + \sum_{j=2}^{k-1} (-1)^j \, \zeta(j) \, a_{k-j}}}{1-k}</math> | |||
where ζ(''s'') is the [[Riemann zeta function]]. For small values, this gives the following values: | |||
{| class = "wikitable" | |||
!k | |||
!<math>a_k</math> | |||
|- | |||
| 1 || 1.0000000000000000000000000000000000000000 | |||
|- | |||
| 2 || 0.5772156649015328606065120900824024310422 | |||
|- | |||
| 3 || −0.6558780715202538810770195151453904812798 | |||
|- | |||
| 4 || −0.0420026350340952355290039348754298187114 | |||
|- | |||
| 5 || 0.1665386113822914895017007951021052357178 | |||
|- | |||
| 6 || −0.0421977345555443367482083012891873913017 | |||
|- | |||
| 7 || −0.0096219715278769735621149216723481989754 | |||
|- | |||
| 8 || 0.0072189432466630995423950103404465727099 | |||
|- | |||
| 9 || −0.0011651675918590651121139710840183886668 | |||
|- | |||
| 10 || −0.0002152416741149509728157299630536478065 | |||
|- | |||
| 11 || 0.0001280502823881161861531986263281643234 | |||
|- | |||
| 12 || −0.0000201348547807882386556893914210218184 | |||
|- | |||
| 13 || −0.0000012504934821426706573453594738330922 | |||
|- | |||
| 14 || 0.0000011330272319816958823741296203307449 | |||
|- | |||
| 15 || −0.0000002056338416977607103450154130020573 | |||
|- | |||
| 16 || 0.0000000061160951044814158178624986828553 | |||
|- | |||
| 17 || 0.0000000050020076444692229300556650480600 | |||
|- | |||
| 18 || −0.0000000011812745704870201445881265654365 | |||
|- | |||
| 19 || 0.0000000001043426711691100510491540332312 | |||
|- | |||
| 20 || 0.0000000000077822634399050712540499373114 | |||
|- | |||
| 21 || −0.0000000000036968056186422057081878158781 | |||
|- | |||
| 22 || 0.0000000000005100370287454475979015481323 | |||
|- | |||
| 23 || −0.0000000000000205832605356650678322242954 | |||
|- | |||
| 24 || −0.0000000000000053481225394230179823700173 | |||
|- | |||
| 25 || 0.0000000000000012267786282382607901588938 | |||
|- | |||
| 26 || −0.0000000000000001181259301697458769513765 | |||
|- | |||
| 27 || 0.0000000000000000011866922547516003325798 | |||
|- | |||
| 28 || 0.0000000000000000014123806553180317815558 | |||
|- | |||
| 29 || −0.0000000000000000002298745684435370206592 | |||
|- | |||
| 30 || 0.0000000000000000000171440632192733743338 | |||
|} | |||
==Asymptotic expansion== | |||
As |''z''| goes to infinity at a constant arg(''z'') we have: | |||
:<math>\ln (1/\Gamma(z)) \sim -z \ln (z) + z + \tfrac{1}{2} \ln \left (\frac{z}{2\pi} \right ) - \frac{1}{12z} + \frac{1}{360z^3} -\frac{1}{1260 z^5}\qquad \qquad \text{for}\quad |\arg(z)| < \pi</math> | |||
==Contour integral representation== | |||
An integral representation due to [[Hermann Hankel]] is | |||
:<math>\frac{1}{\Gamma(z)} = \frac{i}{2\pi} \oint_C (-t)^{-z} e^{-t} \, dt,</math> | |||
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 | |||
:<math>\int_{0}^\infty \frac{1}{\Gamma(x)}\, dx \approx 2.80777024,</math> | |||
which is known as the [[Fransén–Robinson constant]]. | |||
==See also== | |||
* [[Bessel–Clifford function]] | |||
* [[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 | |||
[[Category:Gamma and related functions]] | |||
[[Category:Analytic functions]] |
Revision as of 18:26, 22 January 2014
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 grows no faster than ), but of infinite type (meaning that grows faster than any multiple of |z|, since its growth is approximately proportional to in the left-hand 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.
Taylor series
Taylor series expansion around 0 gives
where γ is the Euler–Mascheroni constant. For k > 2, the coefficient ak for the zk term can be computed recursively as
where ζ(s) is the Riemann zeta function. For small values, this gives the following values:
Asymptotic expansion
As |z| goes to infinity at a constant arg(z) we have:
Contour integral representation
An integral representation due to Hermann Hankel is
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
which is known as the Fransén–Robinson constant.
See also
References
- Thomas Schmelzer & Lloyd N. Trefethen, Computing the Gamma function using contour integrals and rational approximations
- Mette Lund, 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, Gamma Function, MathWorld