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| [[Image:Rgamma plot real.png|thumb|250px|Plot of 1/Γ(x) along the real axis]]
| | Hello and welcome. My title is Numbers Wunder. Doing ceramics is what her family and her appreciate. California is our beginning location. Hiring is her working day job now but she's usually needed her own company.<br><br>Here is my website [http://nuvem.tk/altergalactica/AliceedMaurermy nuvem.tk] |
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| [[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]].]]
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| In [[mathematics]], the '''reciprocal gamma function''' is the [[special function|function]]
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| :<math>f(z) = \frac{1}{\Gamma(z)},</math>
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| 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).
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| 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.
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| [[Karl Weierstrass]] called the reciprocal gamma function the "factorielle" and used it in his development of the [[Weierstrass factorization theorem]].
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| ==Taylor series==
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| [[Taylor series]] expansion around 0 gives
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| :<math>\frac{1}{\Gamma(z)} = z + \gamma z^2 + \left(\frac{\gamma^2}{2} - \frac{\pi^2}{12}\right)z^3 + \cdots</math>
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| 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
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| :<math>a_k = \frac{{- a_2 a_{k-1} + \sum_{j=2}^{k-1} (-1)^j \, \zeta(j) \, a_{k-j}}}{1-k}</math>
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| where ζ(''s'') is the [[Riemann zeta function]]. For small values, this gives the following values:
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| {| class = "wikitable"
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| !k
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| !<math>a_k</math>
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| |-
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| | 1 || 1.0000000000000000000000000000000000000000
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| |-
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| | 2 || 0.5772156649015328606065120900824024310422
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| |-
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| | 3 || −0.6558780715202538810770195151453904812798
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| |-
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| | 4 || −0.0420026350340952355290039348754298187114
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| |-
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| | 5 || 0.1665386113822914895017007951021052357178
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| |-
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| | 6 || −0.0421977345555443367482083012891873913017
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| |-
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| | 7 || −0.0096219715278769735621149216723481989754
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| |-
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| | 8 || 0.0072189432466630995423950103404465727099
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| |-
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| | 9 || −0.0011651675918590651121139710840183886668
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| | 10 || −0.0002152416741149509728157299630536478065
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| | 11 || 0.0001280502823881161861531986263281643234
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| |-
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| | 12 || −0.0000201348547807882386556893914210218184
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| |-
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| | 13 || −0.0000012504934821426706573453594738330922
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| | 14 || 0.0000011330272319816958823741296203307449
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| | 15 || −0.0000002056338416977607103450154130020573
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| | 16 || 0.0000000061160951044814158178624986828553
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| | 17 || 0.0000000050020076444692229300556650480600
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| | 18 || −0.0000000011812745704870201445881265654365
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| |-
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| | 19 || 0.0000000001043426711691100510491540332312
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| | 20 || 0.0000000000077822634399050712540499373114
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| |-
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| | 21 || −0.0000000000036968056186422057081878158781
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| | 22 || 0.0000000000005100370287454475979015481323
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| | 23 || −0.0000000000000205832605356650678322242954
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| | 24 || −0.0000000000000053481225394230179823700173
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| | 25 || 0.0000000000000012267786282382607901588938
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| | 26 || −0.0000000000000001181259301697458769513765
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| | 27 || 0.0000000000000000011866922547516003325798
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| | 28 || 0.0000000000000000014123806553180317815558
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| | 29 || −0.0000000000000000002298745684435370206592
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| | 30 || 0.0000000000000000000171440632192733743338
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| |}
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| ==Asymptotic expansion==
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| As |''z''| goes to infinity at a constant arg(''z'') we have:
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| :<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>
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| ==Contour integral representation==
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| An integral representation due to [[Hermann Hankel]] is
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| :<math>\frac{1}{\Gamma(z)} = \frac{i}{2\pi} \oint_C (-t)^{-z} e^{-t} \, dt,</math>
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| 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.
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| ==Integral along the real axis==
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| Integration of the reciprocal gamma function along the positive real axis gives the value
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| :<math>\int_{0}^\infty \frac{1}{\Gamma(x)}\, dx \approx 2.80777024,</math>
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| which is known as the [[Fransén–Robinson constant]].
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| ==See also==
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| * [[Bessel–Clifford function]]
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| * [[Inverse-gamma distribution]]
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| ==References==
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| * 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]
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| * Mette Lund, [http://www.nbi.dk/~polesen/borel/node14.html An integral for the reciprocal Gamma function]
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| * Milton Abramowitz & Irene A. Stegun, ''[[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]''
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| * [[Eric W. Weisstein]], ''[http://mathworld.wolfram.com/GammaFunction.html Gamma Function]'', MathWorld
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| [[Category:Gamma and related functions]]
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| [[Category:Analytic functions]]
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Hello and welcome. My title is Numbers Wunder. Doing ceramics is what her family and her appreciate. California is our beginning location. Hiring is her working day job now but she's usually needed her own company.
Here is my website nuvem.tk