Gamma Function Calculator

Gamma Function:

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

Gamma Function Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

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 widely used in mathematics, physics, and engineering.

2. How Does the Calculator Work?

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

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

Where:

\( z \) — Input value
\( g \) — Arbitrary constant (7 in our implementation)
\( A_g(z) \) — Series of coefficients

Explanation: The Lanczos approximation provides accurate results for both integer and non-integer values of x.

3. Applications of Gamma Function

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

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is Γ(1)?
A: Γ(1) = 1. This follows from the factorial property since 0! = 1.

Q2: What is Γ(1/2)?
A: Γ(1/2) = √π ≈ 1.77245385091. This is a special value of the Gamma function.

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

Q4: What are the poles of Gamma function?
A: The Gamma function has simple poles at all non-positive integers (0, -1, -2, ...).

Q5: Can Gamma handle complex numbers?
A: While the Gamma function can be extended to complex numbers, this calculator only handles real inputs.