Ters Gama fonksiyonu

Taylor serisi 0 etrafında açılım verir:

${\displaystyle {\frac {1}{\Gamma (z)}}=z+\gamma z^{2}+\left({\frac {\gamma ^{2}}{2}}-{\frac {\pi ^{2}}{12}}\right)z^{3}+\cdots }$

Burada ${\displaystyle \gamma }$ Euler-Mascheroni sabiti'dir.k > 2 için katsayı a_k için z^k terimleri türetilebilir.

${\displaystyle a_{k}=ka_{1}a_{k}-a_{2}a_{k-1}+\sum _{j=2}^{k-1}(-1)^{j}\,\zeta (j)\,a_{k-j}}$

burada ζ(s) Riemann zeta fonksiyonu'dur.

k	${\displaystyle a_{k}}$
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

integral gösterimi Hermann Hankel tarafından;

${\displaystyle {\frac {1}{\Gamma (z)}}={\frac {i}{2\pi }}\oint _{C}(-t)^{-z}e^{-t}dt,}$

Burada C 0 çevresinde pozitif reel eksen etrafında pozitif yönde,artı sonsuza kadar başlar ve biter. Schmelzer & Trefethen'e göre, Hankel integrali Gama fonksiyonunu sayısal değerlendirmesi için en iyi hesaplama yöntemidir.

Ters Gama fonksiyonu'nun pozitif reel eksen etrafında verilen değeri

${\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx\approx 2.80777024,}$

Fransén–Robinson sabiti olarak bilinir..

• Matematiksel fonksiyonların listesi
• Bessel–Clifford function
• Inverse-gamma distribution

• 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 Function6 Temmuz 2017 tarihinde Wayback Machine sitesinde arşivlendi., MathWorld