Gamma Function Calculator

Γ(n) = (n−1)!

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About Gamma Function Calculator

A Gamma function calculator computing Γ(z) = ∫₀^∞ tᶻ⁻¹e⁻ᵗdt. For positive integers: Γ(n) = (n−1)!. Shows key values, reflection formula, and duplication formula. Handles half-integer and real arguments. All calculations are client-side.

Gamma Function Calculator Features

  • Γ(z)
  • ln Γ(z)
  • ψ(z) digamma
  • Key values
  • Properties
Gamma function: Γ(z) = ∫₀^∞ tᶻ⁻¹e⁻ᵗdt. Γ(n) = (n−1)! for positive integers. Γ(1/2) = √π. Recursion: Γ(z+1) = zΓ(z). Reflection: Γ(z)Γ(1−z) = π/sin(πz). Stirling: Γ(z) ≈ √(2π/z)(z/e)ᶻ.

How to Use

Enter a value:

  • z: Input (real number)
  • Γ(z): Gamma function
  • ln Γ(z): Log-gamma

Key Identities

  • Γ(z+1) = zΓ(z)
  • Γ(1/2) = √π
  • Γ(z)Γ(1−z) = π/sin(πz)
  • Γ(z)Γ(z+1/2) = √π·Γ(2z)/2²ᶻ⁻¹

Applications

  • Combinatorics: generalized binomial
  • Probability: normalizing constants
  • Physics: dimensional regularization
  • Number theory: Riemann zeta

Step-by-Step Instructions

  1. 1Enter z value.
  2. 2Get Γ(z).
  3. 3View ln Γ(z).
  4. 4Check identities.
  5. 5See properties.

Gamma Function Calculator — Frequently Asked Questions

Why is Γ(1/2) = √π?+

Γ(1/2) = ∫₀^∞ t⁻¹/²e⁻ᵗdt. Substitute t = u²: = 2∫₀^∞ e⁻ᵘ² du = √π (Gaussian integral). This is the fundamental connection between the Gamma function and the Gaussian distribution.

How is Gamma computed for large z?+

Stirling's approximation: Γ(z) ≈ √(2π/z)(z/e)ᶻ. More precisely: ln Γ(z) ≈ (z−1/2)ln z − z + ln(2π)/2 + 1/(12z) −... Lanczos approximation gives 15-digit accuracy with 7 terms.

Does Gamma have poles?+

Γ(z) has simple poles at z = 0, −1, −2, −3, ... with residues (−1)ⁿ/n!. It never equals zero. The reciprocal 1/Γ(z) is entire (analytic everywhere). The poles correspond to the factorial being undefined for negative integers.

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