Gamma Function Calculator

Calculate values of the Gamma function Γ(z) - the extension of factorial to real and complex numbers. Essential for advanced mathematics, statistics, and engineering applications.

Uses Lanczos approximation for real-number inputs with precision up to 10 decimal places.

Input Value (z)

Supports positive/negative decimals (except non-positive integers)

Result

About the Gamma Function

The Gamma function extends the factorial concept to real and complex numbers, satisfying Γ(n) = (n-1)! for positive integers. It's fundamental in:

Probability distributions (Gamma, Beta, Chi-Squared)
Fractional calculus operations
Physics and engineering applications
Analytic number theory

Calculation Examples

Γ(5) = 4! = 24
Γ(0.5) = √π ≈ 1.77245
Γ(1.7) ≈ 0.908638

Important Properties

Property	Formula
Recurrence	Γ(z+1) = z·Γ(z)
Reflection	Γ(1-z)·Γ(z) = π/sin(πz)
Special Value	Γ(1/2) = √π

Gamma Function Visualization

Range From

Range To

Frequently Asked Questions

What values are undefined for Gamma function?

Γ(z) is undefined for non-positive integers (0, -1, -2, -3, ...) where it has simple poles.

What approximation method is used?

Lanczos approximation with g=7 and 9 coefficients, providing precision ~10^-13 for real inputs.

Can I calculate complex numbers?

This calculator supports real numbers only. Complex Gamma requires specialized algorithms.

Disclaimer

For informational and educational purposes only. Results are approximations and may differ from exact values. Not for use in critical applications.

References

Lanczos Approximation (1964) - Numerical calculation of Gamma functions
Abramowitz & Stegun - Handbook of Mathematical Functions
Artin, Emil - The Gamma Function (1931)

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Gamma Function Calculator - Generalized Factorial

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