Calculate Gamma Function

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 widely used in mathematics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses Lanczos approximation to compute the Gamma function:

\[ \Gamma(z) = \sqrt{2\pi} \left(z + g + \frac{1}{2}\right)^{z+\frac{1}{2}} e^{-\left(z + g + \frac{1}{2}\right)} A(z) \]

Where:

\( z \) — Input value
\( g \) — Lanczos constant (typically 7)
\( A(z) \) — Series approximation

Explanation: The approximation provides high accuracy (about 15 significant digits) for complex numbers with positive real part.

3. Applications of Gamma Function

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

4. Using the Calculator

Tips: Enter any positive real number. The calculator will compute Γ(x). For x ≤ 0, the function uses reflection formula Γ(z)Γ(1-z) = π/sin(πz).

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 are special values of Gamma function?
A: Γ(1/2) = √π, Γ(1) = 1, Γ(3/2) = √π/2, Γ(2) = 1.

Q3: Can Gamma function handle negative numbers?
A: Yes, but it's undefined at non-positive integers (0, -1, -2,...). For other negatives, it uses reflection formula.

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

Q5: What's the difference between Gamma and incomplete Gamma?
A: Incomplete Gamma functions (upper and lower) are variations where the integral has finite bounds instead of 0 to ∞.