📐 Gamma Function Calculator
Calculate Γ(z)
How to Use This Calculator
1
Enter Value of z
Type any real number (except 0 and negative integers)
2
Calculate
See Γ(z), ln(Γ(z)), and related factorial if integer
Gamma Function Definition
Γ(z) = ∫₀∞ tᵗ⁻¹e⁻ᵗ dt, for Re(z) > 0
Key Properties:
- Γ(n + 1) = n! for positive integers n
- Γ(z + 1) = z · Γ(z) (recurrence relation)
- Γ(1/2) = √π
- Γ(z) · Γ(1 - z) = π / sin(πz) (reflection formula)
- Undefined for z = 0, -1, -2, ...
Special Values:
- Γ(1) = 1
- Γ(2) = 1
- Γ(3) = 2
- Γ(4) = 6
- Γ(1/2) ≈ 1.7725
About Gamma Function Calculator
The Gamma Function Calculator computes Γ(z), which extends the factorial function to complex and real numbers. For positive integers, Γ(n + 1) = n!. It's central in calculus, probability, statistics, and physics.
Applications
- Probability: Beta distribution, chi-square distribution
- Physics: Quantum mechanics, statistical mechanics
- Mathematics: Series solutions, integral representations
- Engineering: Signal processing, control theory
Important Notes
- Gamma function is undefined for 0 and negative integers
- For positive integers n, Γ(n) = (n-1)!
- The gamma function is continuous for all real numbers except non-positive integers
Frequently Asked Questions
What is the relationship between Gamma and factorial?
For positive integers n, Γ(n) = (n-1)!. For example, Γ(4) = 3! = 6.
Why is Γ(1/2) = √π?
This follows from the Gaussian integral: ∫₀∞ e⁻ˣ² dx = √π/2, which is Γ(1/2).
Can I calculate gamma for negative numbers?
Yes, except for negative integers (0, -1, -2, ...). Use the reflection formula for negative real numbers.