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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_gamma_function
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gptkbp:appearsIn
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gptkb:chi-squared_distribution
gptkb:gamma_distribution
gptkb:zeta_function
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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:extendsTo
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factorial to complex numbers
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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:hasDuplicationFormula
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Γ(z)Γ(z+1/2) = 2^{1-2z}√π Γ(2z)
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gptkbp:hasIntegralRepresentation
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Γ(z) = ∫₀^∞ t^{z-1} e^{-t} dt
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gptkbp:hasNoZeroes
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true
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gptkbp:hasPoleAt
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non-positive integers
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gptkbp:hasReflectionFormula
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Γ(1-z)Γ(z) = π/sin(πz)
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gptkbp:hasSeriesRepresentation
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gptkb:Weierstrass_product
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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:isAnalyticExceptAt
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non-positive integers
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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:relatedTo
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gptkb:digamma_function
gptkb:polygamma_function
beta function
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gptkbp:satisfies
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Γ(n) = (n-1)!
Γ(z+1) = zΓ(z)
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gptkbp:solvedBy
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functional equation Γ(z+1) = zΓ(z)
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gptkbp:usedFor
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number theory
solving differential equations
combinatorics
analytic continuation
defining special functions
evaluating integrals
representation of probability distributions
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gptkbp:usedIn
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gptkb:probability_theory
complex analysis
physics
statistics
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gptkbp:bfsParent
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gptkb:Gamma
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gptkbp:bfsLayer
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5
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