1. What is the Gamma Function?

The Gamma function (Γ) is a generalization of the factorial function to complex and real numbers. 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 the Gamma function:

Key Points:

• For x > 0, the integral converges
• For positive integers n, Γ(n) = (n-1)!
• Γ(1/2) = √π
• The function extends to complex numbers (except non-positive integers)

3. Importance of Gamma Function

Applications: The Gamma function appears in probability distributions (Gamma, Chi-squared, Student's t), fractional calculus, analytic number theory, and physics (quantum mechanics, statistical mechanics).

4. Using the Calculator

Tips: Enter a positive real number (x > 0). The calculator will compute Γ(x) using a precise approximation method.

5. Frequently Asked Questions (FAQ)

Q1: How is Gamma function related to factorial?
A: For positive integers n, Γ(n) = (n-1)!. So Γ(5) = 4! = 24.

Q2: What's the Gamma function of 1/2?
A: Γ(1/2) = √π ≈ 1.77245385091.

Q3: Can Gamma function handle negative numbers?
A: The Gamma function is defined for all complex numbers except non-positive integers (0, -1, -2, ...).

Q4: Why use Lanczos approximation?
A: It provides excellent accuracy (about 15 decimal digits) for positive real numbers.

Q5: How do calculators compute Gamma function?
A: Most use approximation methods like Lanczos or Stirling's approximation for efficiency.