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Article 11 — Appendix A.33Γ or Gamma gamma functionCategory. Mathematics. Abstract. Gamma function: definition, graph, properties and identities. Reference. This article is a part of Librow professional formula calculator project. See also. Special case of integer numbers — factorial n!. Limited offerProfessional Librow Calculatorvisitfor free
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7.4 MB for Windows 1. DefinitionGamma function is defined as integral ![]() or ![]() 2. PlotGamma function defined everywhere on real axis, except its singular points n = 0, −1, −2, ... — so, function domain is ...∪(−2, −1)∪(−1, 0)∪(0, +∞). Its plot is depicted below — fig. 1. ![]() Function codomain is entire real axis except 0: (−∞, 0)∪(0, +∞). 3. IdentitiesConnection to factorial: Γ(n) = (n − 1)!Factorial-like properties: Γ(x + 1) = x Γ(x) Γ(1 − x) = −x Γ(−x)Extension to negative half-axis: Γ(1 − x) = π/[Γ(x) sin(πx)]Doulbe argument: Γ(2x) = (2π)−1/2 22x − 1/2 Γ(x) Γ(x + 1/2)Triple argument: Γ(3x) = (2π)−1 33x − 1/2 Γ(x) Γ(x + 1/3) Γ(x + 2/3)Quadruple argument: Γ(4x) = (2π)−3/2 44x − 1/2 Γ(x) Γ(x + 1/4) Γ(x + 1/2) Γ(x + 3/4)Genaral formula for multiple argument: Γ(nx) = (2π)(1−n)/2 nnx − 1/2 Γ(x) Γ(x + 1/n) ... Γ(x + (n − 1)/n)Half-integer argument: Γ(−5 /2) = −8 /15 √πΓ(−3 /2) = 4 /3 √π Γ(−1 /2) = −2 √π Γ(1 /2) = √π Γ(3 /2) = 1 /2 √π Γ(5 /2) = 3 /4 √π and in general: Γ(1/2 + n) = (2n − 1)!! /2n√π = 1 × 3 × 5 × ... × (2n − 1) /2n√πfor negative values: Γ(1/2 − n) = (−1)n2n/(2n − 1)!! √π = (−1)n2n/[1 × 3 × 5 × ... × (2n − 1)] √πas well, for positive odd n: Γ(n /2) = (n − 2)!! /2(n − 1)/2 √πand for negative odd n: Γ(n /2) = (−1)(n + 1)/2 2(n + 1)/2/n!! √π4. SupportGamma function Γ or Gamma of the complex argument is supported by professional version of the Librow calculator. 5. How to useTo calculate gamma function of the number:
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To calculate gamma function of the current result:
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To calculate gamma function of the number x in memory:
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