Gamma Function Calculator

Calculate the Gamma function using the Lanczos approximation. For positive integers, the gamma function equals (n-1) factorial.

Enter Value

Note: The Gamma function is undefined for non-positive integers (0, -1, -2, ...). For positive integers n, the result equals (n-1)!

Special Values (click to calculate)

xGamma(x)Note
1/21.7724539sqrt(pi)
110! = 1
3/20.8862269sqrt(pi)/2
211! = 1
322! = 2
463! = 6
5244! = 24
61205! = 120
-1/2-3.5449077-2*sqrt(pi)
-3/22.36327184*sqrt(pi)/3

Result

Gamma(x)
--
Input x --
Is positive integer? --
Factorial form --
ln(Gamma(x)) --
1 / Gamma(x) --

Step-by-Step Solution

Gamma(n) = (n-1)! for positive integers

Understanding the Gamma Function

The Gamma function, denoted by the Greek letter capital gamma, is one of the most important special functions in mathematics. It extends the factorial function to complex and real numbers. For positive integers n, the relationship is simply: Gamma(n) = (n-1)!

The Gamma function is defined by the integral: Gamma(x) = integral from 0 to infinity of t^(x-1) * e^(-t) dt, which converges for all complex numbers with positive real part. Through analytic continuation, it can be extended to all complex numbers except the non-positive integers.

Key Properties

Recurrence Relation

The fundamental recursive property linking consecutive values.

Gamma(x+1) = x * Gamma(x)

Reflection Formula

Euler's reflection formula connects values at x and 1-x.

Gamma(x)Gamma(1-x) = pi/sin(pi*x)

Duplication Formula

Legendre's duplication formula relates Gamma(x) to Gamma(2x).

Gamma(x)Gamma(x+1/2) = sqrt(pi)/2^(2x-1) * Gamma(2x)

Half-Integer Values

At half-integers, the Gamma function involves sqrt(pi).

Gamma(1/2) = sqrt(pi) = 1.7724539...

The Lanczos Approximation

This calculator uses the Lanczos approximation, a practical method for computing the Gamma function numerically. The approximation is based on a series expansion that converges rapidly, providing about 15 digits of precision with just a few terms. It is widely used in scientific computing libraries.

The approximation takes the form: Gamma(z+1) = sqrt(2*pi) * (z + g + 0.5)^(z+0.5) * exp(-(z+g+0.5)) * A_g(z), where g is a constant and A_g(z) is a series that converges very quickly.

Applications of the Gamma Function

  • Probability & Statistics: The Gamma, Chi-squared, Student's t, and F distributions all involve the Gamma function.
  • Physics: Appears in quantum mechanics, statistical mechanics, and string theory.
  • Number Theory: Connected to the Riemann zeta function and prime number theory.
  • Engineering: Used in signal processing, control theory, and reliability analysis.
  • Combinatorics: Generalizes factorials, binomial coefficients, and multinomials to real/complex arguments.

Common Pitfalls

  • Gamma(0) is undefined (it has a pole at 0).
  • Gamma function has poles at all non-positive integers: 0, -1, -2, -3, ...
  • Gamma grows extremely fast: Gamma(171) already overflows 64-bit floating point.
  • For large arguments, use log-gamma (ln Gamma) to avoid overflow.