Calculate Gamma Function
Online calculator and formulas for calculating the Euler Gamma function
Gamma Function Calculator
Using the Calculator
Enter the argument x and click 'Calculate'. For integers ≤ 0 the result is ±∞.
Gamma Function Curve
Mouse pointer on the graph shows the values.
Y-scale is limited to ±20 for better visualization.
Euler Gamma Function Formulas
Integral Representation
for Re(x) > 0
Factorial Relationship
for natural numbers n
Recurrence Formula
Reflection Formula
Special Values
Important Values
Half-Integer Values
≈ 1.772453851
Example
Γ(0.5) = √π ≈ 1.772
Γ(1.5) = 0.5 × √π ≈ 0.886
Γ(2.5) = 1.5 × 0.5 × √π ≈ 1.329
Comprehensive Description of the Gamma Function
Mathematical Definition
The Euler Gamma function is one of the most important special functions in mathematics. It extends the factorial function to real and complex numbers and is denoted by the Greek symbol Γ (Gamma).
Euler Gamma Function
The Γ(x) function is one of the most important special functions in analysis and function theory. It generalizes the factorial to real numbers.
Historical Background
The Gamma function was introduced by Leonhard Euler in the 18th century and later developed further by Adrien-Marie Legendre and Carl Friedrich Gauss. It plays a central role in many areas of mathematics.
Properties and Applications
Mathematical Applications
- Probability theory (Beta distribution, Gamma distribution)
- Combinatorics (generalization of factorial)
- Analytic number theory (Riemann zeta function)
- Differential equations and integral calculus
Physical Applications
- Quantum mechanics (hydrogen atom, harmonic oscillator)
- Statistical mechanics (Maxwell-Boltzmann distribution)
- Nuclear physics (radioactive decay)
- Astrophysics (stellar evolution)
Special Properties
- Holomorphic: Γ(z) is holomorphic except at z = 0, -1, -2, ...
- Functional equation: Γ(z+1) = z·Γ(z)
- Convexity: log Γ(x) is convex for x > 0
- Stirling's formula: Asymptotic expansion for large x
Interesting Facts
- Γ(x) has poles at x = 0, -1, -2, -3, ...
- The Bohr-Mollerup theorem characterizes Γ uniquely
- Connection to Beta function: B(x,y) = Γ(x)Γ(y)/Γ(x+y)
- The Gamma function is the unique log-convex extension of the factorial
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