Gamma Function Calculator

Gamma Function Definition:

\[ \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt \]

Gamma Function Value:

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1. What is the Gamma Function?

The Gamma function (Γ) is a generalization of the factorial function to complex and real number arguments. 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 Lanczos approximation to compute the Gamma function:

\[ \Gamma(z) = \sqrt{2\pi} \left(z + g + \frac{1}{2}\right)^{z+\frac{1}{2}} e^{-\left(z+g+\frac{1}{2}\right)} \left[c_0 + \frac{c_1}{z+1} + \frac{c_2}{z+2} + \cdots + \frac{c_n}{z+n}\right] \]

Where:

\( z \) — Input value
\( g \) — Lanczos constant (typically 7)
\( c_0, c_1, \ldots, c_n \) — Precomputed coefficients

Explanation: The Lanczos approximation provides high accuracy for the Gamma function calculation, especially 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 physics. It's essential in statistical mechanics and quantum field theory.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between Gamma and factorial?
A: For positive integers n, Γ(n) = (n-1)!. The Gamma function extends factorial to all complex numbers except negative integers.

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

Q3: Can Gamma be calculated for negative numbers?
A: The Gamma function is defined for all complex numbers except non-positive integers (0, -1, -2, ...). For negative non-integers, it can be calculated using reflection formula.

Q4: What's the accuracy of this calculator?
A: The Lanczos approximation provides about 15 decimal digits of accuracy for positive real numbers.

Q5: Where is Gamma function used in real applications?
A: It's used in probability distributions (Gamma, Beta, Chi-squared), physics (quantum mechanics, statistical mechanics), and engineering (signal processing).