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 integral definition of the Gamma function:

For practical computation, it implements the Lanczos approximation, which provides accurate results for positive real numbers.

3. Importance of Gamma Function

Details: The Gamma function appears in many areas of mathematics including probability distributions (gamma, chi-squared), complex analysis, and number theory. It's essential for extending factorial concepts to non-integer values.

4. Using the Calculator

Tips: Enter any positive real number (x > 0). The calculator will compute Γ(x) using numerical approximation methods. For integers, the result will be (x-1) factorial.

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: Can Gamma be calculated for negative numbers?
A: Yes, except for negative integers (where it has simple poles). The calculator handles positive numbers only.

Q3: What's the Gamma of 0.5?
A: Γ(0.5) = √π ≈ 1.77245385091. This is a special case important in probability theory.

Q4: How accurate is this calculator?
A: The Lanczos approximation provides about 15 digits of accuracy for most values.

Q5: What are some practical applications?
A: Used in probability distributions, physics (quantum mechanics), engineering (signal processing), and more.