1. What is the Gamma Function?

The Gamma function 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 Gamma function definition:

Where:

• \( x \) — Input value (must be positive)
• \( t \) — Integration variable

Explanation: The Gamma function extends the factorial concept to non-integer values through this integral definition.

3. Importance of Gamma Function

Details: The Gamma function appears in probability distributions (Gamma, Chi-squared), fractional calculus, analytic number theory, and physics problems.

4. Using the Calculator

Tips: Enter a positive real number. The calculator will compute an approximation of the Gamma function at that point.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between Gamma and factorial?
A: For positive integers n, Γ(n) = (n-1)!. For example, Γ(5) = 4! = 24.

Q2: What is Γ(1/2)?
A: Γ(1/2) = √π ≈ 1.77245.

Q3: Can Gamma be calculated for negative numbers?
A: Yes, but not for negative integers (where it has simple poles). For other negatives, use Γ(x) = Γ(x+1)/x.

Q4: What are some applications of Gamma function?
A: Used in probability distributions, physics, fractional calculus, and solving certain integrals.

Q5: How is the calculation performed?
A: This calculator uses a simple recursive approximation. More sophisticated methods like Lanczos approximation would be used in practice.