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:

Implementation: The Lanczos approximation provides good accuracy for positive real numbers with a relative error of less than 2×10⁻¹⁰.

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 quantum physics.

4. Using the Calculator

Tips: Enter any positive real number. The calculator will compute Γ(x). For x ≤ 0, the function is either undefined or infinite.

5. Frequently Asked Questions (FAQ)

Q1: How is Gamma related to factorial?
A: For positive integers n, Γ(n) = (n-1)!. Γ(1) = 1, Γ(2) = 1, Γ(3) = 2, Γ(4) = 6, etc.

Q2: What is Γ(1/2)?
A: Γ(1/2) = √π ≈ 1.77245385091

Q3: Can Gamma handle non-integer values?
A: Yes, Γ(x) is defined for all complex numbers except non-positive integers.

Q4: Why use Lanczos approximation?
A: It provides excellent accuracy with relatively simple computation compared to other methods.

Q5: How is this implemented in calculators?
A: Most calculators use similar approximation methods due to the complexity of exact calculation.