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:<math>\Gamma(z+1)=z\Gamma(z)\,.</math>
Sabab Γ(1) = 1, thisdina relationkaitan impliesieu thatngakibatkeun yen
:<math>\Gamma(n+1)=n!\,</math>
forkeur allsakabeh [[natural number|naturalwilangan numbersnatural]] ''n''. ItIeu canbisa furtherdipake bekeur used to extendngalegaan Γ(''z'') to ajadi [[meromorphic function|fungsi meromorpik]] defineddiartikeun forkeur allsakabar complexwilangan numberskompleks ''z'' exceptial ''z'' = 0, −1, −2, −3, ... byku [[analytic continuation|analisa kontinyu]].
Hal nu leuwih lega ilaharna dumasar salaku fungsi gamma.
It is this extended version that is commonly referred to as the gamma function.
Notasi alternatip nu kadangkala dipake nyaeta '''fungsi Pi''', nu dina watesan fungsi gamma nyaeta
An alternative notation which is sometimes used is the '''Pi function''', which in terms of the gamma function is
:<math>\Pi(z) = \Gamma(z+1) = z\Gamma(z).</math>
Kadangkala oge manggihkeun
We also sometimes find
:<math>\pi(z) = {1 \over \Pi(z)}\,</math>
whichnu isngarupakeun anhiji [[entire function|fungsi sakabehna]], defineddiartikeun forkeur everysakabeh complexwilangan numberkompleks. ThatYen π(''z'') isngarupakeun entiresakabeh nu entailsdiperlukeun itanu hasteu nomibanda poleskutub, somangka Γ(''z'') hasteu nomibanda [[zero|nol]]s.
PerhapsBisa theoge mostnilai well-knownkeur value of thefungsi gamma function at adina non-integer isnyaeta
:<math>\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.</math>
TheFungsi gamma functionmibanda has ahiji [[pole (complex analysis)|polekutub]] of orderorde 1 atdina ''z'' = −''n'' forkeuw everysakabeh [[natural number|wilangan alami]] ''n''; the [[residue (complex analysis)|residuesesana]] there is givendiberekeun byku
:<math>\operatorname{Res}(\Gamma,-n)=\frac{(-1)^n}{n!}.</math>
TheBentuk followingkakali multiplicative form of thefungsi gamma functionsaterusna isnyaeta valid forkeur allsakabeh complexwilangan numberskompleks ''z'' whichnu arelain notinteger non-positive integerspositip:
:<math>\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}</math>
wherenumana γ is thengarupakeun [[Euler-Mascheroni constant|konstanta Euler-Mascheroni]].
The [[Bohr-Mollerup theorem|TeoremaBohr-Mollerup]] statesnangtukeun thatyen amongantara allsakabeh functionsfungsi extendingdilegaan theku factorialfungsi functionsfaktorial tokana thewilangan positiveriil real numberspositip, onlyngan thelamun gammafungsi functiongamma isngarupakeun log-convex.
== Kaitan jeung fungsi sejen ==
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