Gamma Function Calculator

Gamma Function:

\[ \Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt \]

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 the Lanczos approximation to compute the Gamma function:

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

Where:

\( z \) — Input value
\( g \) — Arbitrary parameter (typically 7)
\( A_g(z) \) — Series expansion with carefully chosen coefficients

Explanation: The Lanczos approximation provides accurate results for the Gamma function across the positive real numbers.

3. Importance of Gamma Function

Details: The Gamma function appears in many areas of mathematics including probability distributions (Gamma, Chi-squared, Student's t), number theory, and the solution of certain differential equations.

4. Using the Calculator

Tips: Enter any positive real number (x > 0). The calculator will compute Γ(x). For negative numbers, the Gamma function is defined via analytic continuation.

5. Frequently Asked Questions (FAQ)

Q1: What is Γ(1)?
A: Γ(1) = 1, as it's equivalent to 0! (0 factorial).

Q2: What is Γ(1/2)?
A: Γ(1/2) = √π ≈ 1.77245385091.

Q3: How is Gamma related to factorial?
A: For positive integers n, Γ(n) = (n-1)!. For example, Γ(5) = 4! = 24.

Q4: What about negative numbers?
A: The Gamma function has simple poles at non-positive integers. For other negative numbers, it can be computed using Γ(x) = π/(Γ(1-x)sin(πx)).

Q5: Why is Gamma function important?
A: It appears in many probability distributions, complex analysis, and as solutions to important differential equations in physics.