Gamma Function Calculator
Calculate the gamma function Γ(z) for real numbers with step-by-step solutions
Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt
For integers: Γ(n) = (n-1)! | Special: Γ(1/2) = √π ≈ 1.7724538509
Note: Γ(z) is undefined at z = 0, -1, -2, ... Uses Γ(z+1) = z! for any real number z
Quick Examples
Undefined Value (Pole)
The Gamma function has a pole at z = . It is undefined at 0 and negative integers.
Γ() !
= ()! =
Reciprocal 1/Γ(z)
Log Gamma ln(Γ(z))
Input Type
Step-by-Step Solution
Identify the input value
z =
Apply the Gamma function Γ(z) Apply generalized factorial z! = Γ(z+1)
For integer z: Γ() = ()! Using Lanczos approximation for numerical computation
Result
Γ() = ! =
Special Values Reference
| z | Γ(z) | Equivalent |
|---|---|---|
| 1 | 1 | 0! = 1 |
| 2 | 1 | 1! = 1 |
| 3 | 2 | 2! = 2 |
| 1/2 | √π ≈ 1.7724538 | Special value |
| 3/2 | √π/2 ≈ 0.8862269 | (1/2)! = (1/2)√π |
| -1/2 | -2√π ≈ -3.5449077 | Reflection formula |
| 0, -1, -2, ... | Undefined | Poles of Γ(z) |
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About Gamma Function Calculator
What is the Gamma Function?
The Gamma function Γ(z) is a fundamental mathematical function that extends the factorial function to complex and real numbers. It is defined for all complex numbers except non-positive integers.
Mathematical Definition
For complex numbers with positive real part (ℜ(z) > 0), the Gamma function is defined by the integral:
Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt
Relation to Factorial
For any positive integer n:
Γ(n) = (n-1)!
Or equivalently: Γ(n+1) = n!
Examples:
- Γ(1) = 0! = 1
- Γ(2) = 1! = 1
- Γ(3) = 2! = 2
- Γ(4) = 3! = 6
- Γ(5) = 4! = 24
Key Properties
Recursive Property
Γ(z+1) = z × Γ(z)
This allows computing Gamma values for any real or complex number.
Special Values
- Γ(1/2) = √π ≈ 1.7724538509
- Γ(3/2) = (1/2)√π ≈ 0.8862269255
- Γ(-1/2) = -2√π ≈ -3.5449077018
Euler's Reflection Formula
For non-integer values of z:
Γ(z) × Γ(1-z) = π / sin(πz)
Poles of the Gamma Function
The Gamma function has simple poles (undefined points) at:
- z = 0, -1, -2, -3, ... (all non-positive integers)
Applications
- Probability and Statistics - Appears in gamma, beta, chi-squared distributions
- Complex Analysis - Fundamental in analytic number theory
- Physics - Quantum mechanics, statistical mechanics
- Combinatorics - Generalized binomial coefficients
- Engineering - Signal processing, control theory
Stirling's Approximation
For large values of z:
Γ(z) ≈ √(2π/z) × (z/e)^z
Common Mistakes
- Confusing Γ(n) with n! - Remember: Γ(n) = (n-1)!, not n!
- Using non-positive integers - Γ(z) is undefined at z = 0, -1, -2, ...
- Forgetting the shift - The gamma function is shifted by 1 from factorial