Enter the real and imaginary parts, then click Calculate Gamma.

The Gamma function Γ(z) is a mathematical extension of the factorial function to complex numbers. For positive integers, Γ(n) = (n−1)! . For all complex numbers except non-positive integers, the Gamma function is defined by the integral Γ(z) = ∫₀^∞ t^(z−1) e^(−t) dt. The function has several important properties: Γ(z+1) = z·Γ(z) (the recurrence relation), Γ(1/2) = √π, and the reflection formula Γ(z)·Γ(1−z) = π/sin(πz). The Gamma function appears extensively in probability theory (in the gamma distribution), statistical physics, number theory, and complex analysis. This calculator uses the Lanczos approximation, which provides high accuracy across the complex plane. It handles real and complex arguments, with special handling for integer values and the reflection formula for negative arguments. The result is displayed as a complex number with magnitude and phase.

1. Enter the Real Part (Re z) of the argument.
2. Enter the Imaginary Part (Im z) of the argument (use 0 for real numbers).
3. Click Calculate Gamma to evaluate the Gamma function.
4. The result shows the complex value, magnitude, and phase.
5. For positive integers, the factorial value is shown as verification.
6. Click Reset to restore default example values.