1. What is the Gamma Function?

The gamma function (Γ) is a generalization of the factorial function to complex and real number arguments. For positive integers, Γ(n) = (n-1)!. It's widely used in mathematics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses the Lanczos approximation to compute:

Key Properties:

• Γ(1) = 1
• Γ(n) = (n-1)! for positive integers n
• Γ(x+1) = xΓ(x) (Recurrence relation)
• Γ(1/2) = √π

3. Applications of Gamma Function

Details: The gamma function appears in probability distributions (gamma, chi-squared, Student's t), fractional calculus, analytic number theory, and quantum physics.

4. Using the Calculator

Tips: Enter any positive real number. The calculator will compute Γ(x) using high-precision approximation.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between gamma and factorial?
A: Gamma extends factorial to non-integer values: Γ(n) = (n-1)! for positive integers.

Q2: Can gamma handle negative numbers?
A: Γ(x) is defined for all complex numbers except non-positive integers (0, -1, -2,...).

Q3: What's the precision of this calculator?
A: It uses Lanczos approximation with 7-digit precision for display.

Q4: What are common gamma function values?
A: Γ(1/2)=√π≈1.772454, Γ(1)=1, Γ(2)=1, Γ(3)=2, Γ(4)=6, etc.

Q5: How is gamma related to beta function?
A: B(x,y) = Γ(x)Γ(y)/Γ(x+y), where B is the beta function.