1. What is the Gamma Function?

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

2. How Does the Calculator Work?

The calculator uses the Lanczos approximation to compute the Gamma function:

Where:

• \( z \) — Complex number input (real part > 0 in this implementation)
• \( t \) — Integration variable

Explanation: The Gamma function extends the factorial concept to real and complex numbers, with applications across mathematics, physics, and engineering.

3. Importance of Gamma Function

Details: The Gamma function is fundamental in many areas of mathematics including complex analysis, probability theory, and combinatorics. It appears in solutions to differential equations and is used in defining various probability distributions.

4. Using the Calculator

Tips: Enter a positive real number for z. The calculator will compute the Gamma function value using numerical approximation methods.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between Gamma and factorial?
A: For positive integers n, Γ(n) = (n-1)!. This makes it a generalization of the factorial function.

Q2: What are special values of the Gamma function?
A: Γ(1/2) = √π, Γ(1) = 1, Γ(3/2) = √π/2, and Γ(n) = (n-1)! for positive integers n.

Q3: Can the Gamma function handle negative numbers?
A: The Gamma function is defined for all complex numbers except non-positive integers (0, -1, -2, -3, ...).

Q4: What is the Stirling approximation?
A: For large |z|, Γ(z) ≈ √(2π) z^(z-1/2) e^(-z), which is useful for asymptotic analysis.

Q5: Where is the Gamma function used in real applications?
A: It appears in probability distributions (gamma, chi-squared, Student's t), physics (quantum mechanics, statistical mechanics), and engineering (signal processing).