Gamma 1. Gamma Distribution Function The gamma function is represented by Γ(y) If α is a positive real number then Γ(α) is defined as Γ(α) = 0 ∫∞ ( y a1 ey dy) for α > 0 If α = 1 Γ(1) = 0 ∫∞ (ey dy) = 1 If we change the variable to y = λz we can use this definition for gamma distribution Γ(α) = 0.
and so Γ ( n + 1) / n! = 1 for all n To make this rigorous would require justifying two key steps the interchange ∑ ∫ = ∫ ∑ and the conclusion that ∑ a n x n = ∑ b n x n for all | x | < 1 implies a n = b n for all n Share Follow this answer to receive notifications answered Jan 19 ’18 at 1216 Barry Cipra.
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Gamma function proof of gamma $\\;Γ(1/2) = \\sqrt \\pi\\;$
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