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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:alsoKnownAs
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gptkb:Euler's_integral_of_the_second_kind
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gptkbp:codomain
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gptkb:Complex_numbers
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gptkbp:definedIn
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Γ(z) = ∫₀^∞ t^{z-1} e^{-t} dt, Re(z) > 0
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gptkbp:domain
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Complex numbers except non-positive integers
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gptkbp:generalizes
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Factorial function
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gptkbp:hasAsymptoticExpansion
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gptkb:Stirling's_approximation
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gptkbp:hasPoleAt
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Non-positive integers
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gptkbp:hasResidueAt
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z = -n, residue = (-1)^n/n! for n=0,1,2,...
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gptkbp:hasSpecialCase
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gptkb:Mellin_transform
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gptkbp:hasZeros
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nan
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https://www.w3.org/2000/01/rdf-schema#label
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Gamma function
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gptkbp:introduced
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gptkb:Leonhard_Euler
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gptkbp:introducedIn
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1729
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gptkbp:isAnalytic
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Everywhere except non-positive integers
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gptkbp:isEntire
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False
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gptkbp:isLogConvex
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True
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gptkbp:isMeromorphic
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True
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gptkbp:isSpecialFunction
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True
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gptkbp:multiplicationTheorem
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Γ(nz) = (2π)^{(1-n)/2} n^{nz-1/2} ∏_{k=0}^{n-1} Γ(z + k/n)
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gptkbp:property
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Γ(z+1) = zΓ(z)
Γ(1) = 1
Γ(1/2) = √π
Γ(n) = (n-1)! for positive integers n
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gptkbp:reflectionFormula
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Γ(1-z)Γ(z) = π/sin(πz)
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gptkbp:relatedTo
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gptkb:Beta_function
gptkb:Digamma_function
gptkb:Polygamma_function
Incomplete gamma function
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gptkbp:satisfies
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Multiplication theorem
Reflection formula
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gptkbp:stirlingApproximation
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Γ(z) ~ √(2π) z^{z-1/2} e^{-z} as z→∞
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gptkbp:usedIn
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gptkb:Physics
gptkb:Probability_theory
Engineering
Statistics
Complex analysis
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gptkbp:bfsParent
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gptkb:Riemann_zeta_function
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gptkbp:bfsLayer
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5
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