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 defined for all complex numbers except non-positive integers.

2. How Does the Calculator Work?

The calculator computes the Gamma function using the integral definition:

Where:

• \( x \) — Input value (must be positive)
• \( t \) — Integration variable
• \( e \) — Euler's number (~2.71828)

Explanation: The Gamma function extends the factorial concept to real and complex numbers, with applications in probability, statistics, and physics.

3. Importance of Gamma Function

Details: The Gamma function is fundamental in many areas of mathematics, including probability distributions (Gamma, Beta, Chi-squared), number theory, and physics.

4. Using the Calculator

Tips: Enter any positive real number. The calculator will compute Γ(x). For integers, Γ(n) = (n-1)!.

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 function handle non-integer values?
A: Yes, Γ(x) is defined for all complex numbers except non-positive integers (0, -1, -2, ...).

Q3: What are some special values of Gamma function?
A: Γ(1/2) = √π, Γ(1) = 1, Γ(2) = 1. The function has poles at non-positive integers.

Q4: How is Gamma function computed numerically?
A: Common methods include Lanczos approximation, Stirling's formula, and continued fractions.

Q5: Where is Gamma function used in real applications?
A: It appears in probability distributions, physics (quantum mechanics), engineering, and fractional calculus.