Gamma Function Calculator
Calculate gamma function values with high precision for real and complex numbers
Calculate Gamma Function
Argument Type
Examples: 5, 2.5, -1.5, 0.5
Gamma Function Result
Step-by-Step Solution
Common Gamma Function Values
Positive Integers
Γ(1) = 0! = 1
Γ(2) = 1! = 1
Γ(3) = 2! = 2
Γ(4) = 3! = 6
Γ(5) = 4! = 24
Half-Integers
Γ(1/2) = √π ≈ 1.7725
Γ(3/2) = (1/2)√π ≈ 0.8862
Γ(5/2) = (3/4)√π ≈ 1.3293
Γ(7/2) = (15/8)√π ≈ 3.3234
Negative Half-Integers
Γ(-1/2) = -2√π ≈ -3.5449
Γ(-3/2) = (4/3)√π ≈ 2.3633
Γ(-5/2) = -(8/15)√π ≈ -0.9453
Key Properties
Factorial Relation
Γ(n) = (n-1)! for positive integers
Functional Equation
Γ(z+1) = z × Γ(z)
Special Value
Γ(1/2) = √π
Reflection Formula
Γ(z)Γ(1-z) = π/sin(πz)
Quick Tips
Understanding the Gamma Function
What is the Gamma Function?
The Gamma function Γ(z) is a generalization of the factorial function to real and complex numbers. It extends the concept of n! to non-integer values, providing a continuous interpolation of the factorial.
Mathematical Definition
For complex numbers with positive real parts, the Gamma function is defined by the integral:
Applications
Statistics & Probability
Used in gamma distribution, beta function, and various statistical models
Physics
Appears in quantum mechanics, thermodynamics, and special relativity
Number Theory
Connected to Riemann zeta function and analytic number theory
Approximation Methods
Lanczos Approximation
High accuracy for moderate values
Stirling's Approximation
Efficient for large arguments
Nemes' Approximation
Simple formula with good accuracy