1. What is the Gamma Function?

The Gamma function is an extension of the factorial function to complex numbers, with the argument shifted down by 1. For positive integers, Γ(n) = (n-1)!. It's defined for all complex numbers except non-positive integers.

2. How Does the Calculator Work?

The calculator uses the Gamma function definition:

For computational purposes, this implementation uses the Lanczos approximation which provides accurate results for real numbers greater than 0.

3. Importance of Gamma Function

Details: The Gamma function is fundamental in various areas of mathematics including combinatorics, probability theory, and complex analysis. It appears in solutions to differential equations and is used in defining other important functions like the Beta function.

4. Using the Calculator

Tips: Enter a positive real number for z. The calculator will compute Γ(z) using numerical approximation methods. Note that for integer values, Γ(n) = (n-1)!.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between Gamma and factorial?
A: For positive integers n, Γ(n) = (n-1)!. So Γ(5) = 4! = 24, Γ(6) = 5! = 120, etc.

Q2: Can Gamma function handle non-integer values?
A: Yes, that's one of its main purposes. Γ(1.5) = √π/2 ≈ 0.8862269255, Γ(2.5) = 3√π/4 ≈ 1.3293403882.

Q3: What are the special values of Gamma function?
A: Γ(1) = 1, Γ(1/2) = √π, Γ(n) = ∞ for n = 0, -1, -2, -3, ...

Q4: What is the range of valid inputs?
A: This calculator accepts positive real numbers. The Gamma function is defined for all complex numbers except non-positive integers.

Q5: How accurate is the approximation?
A: The Lanczos approximation used here provides high accuracy (typically within 10-15 decimal places) for positive real arguments.