Calculate Γ(x), log Γ(x), digamma ψ(x), and reciprocal gamma for any real number. Explore special values, reference tables, and magnitude visualizations.
The gamma function Γ(x) extends the factorial to all real (and complex) numbers. For positive integers, Γ(n) = (n−1)!, but the function is defined continuously for all real x except the non-positive integers where it has poles. Discovered by Euler and refined by Legendre, the gamma function appears throughout mathematics — from probability distributions (the chi-squared, Student-t, and F distributions all involve Γ) to physics (quantum mechanics, string theory) and engineering (signal processing, control theory). Our calculator uses the Lanczos approximation, which provides excellent accuracy across the real line. Enter any real number and instantly see Γ(x), its natural logarithm log Γ(x) (essential for numerical stability with large arguments), the digamma function ψ(x) = d/dx ln Γ(x), and the reciprocal 1/Γ(x) which is an entire function with no poles. Special value presets let you verify famous results like Γ(1/2) = √π, and the configurable table generates a sweep of gamma values across any interval. The magnitude bar chart provides a visual feel for how rapidly the gamma function grows — faster than exponential for large positive arguments — making it one of the fastest-growing functions encountered in practice.
Gamma Function Calculator helps you solve gamma function problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Input Value (x), Decimal Places, Start once and immediately inspect log Γ(x), ψ(x) Digamma, 1/Γ(x) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Γ(x) = ∫₀^∞ t^(x−1) e^(−t) dt. Recurrence: Γ(x+1) = x·Γ(x). For positive integers: Γ(n) = (n−1)!. Reflection: Γ(x)·Γ(1−x) = π / sin(πx).
Result: log Γ(x) shown by the calculator
Using the preset "Γ(1) = 1", the calculator evaluates the gamma function setup, applies the selected algebra rules, and reports log Γ(x) with supporting checks so you can verify each transformation.
This calculator takes Input Value (x), Decimal Places, Start, End and applies the relevant gamma function relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use log Γ(x), ψ(x) Digamma, 1/Γ(x), |Γ(x)| to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Γ(x) is the continuous extension of the factorial function to all real and complex numbers. For positive integers n, Γ(n) = (n−1)!.
This is a historical convention from Legendre. Some authors use Π(x) = Γ(x+1) = x! to avoid the off-by-one shift.
The digamma function ψ(x) = d/dx ln Γ(x) = Γ′(x)/Γ(x) is the logarithmic derivative of the gamma function. It appears in many optimization and statistics formulas.
The gamma function has simple poles (goes to ±∞) at all non-positive integers. It is undefined at these points.
The Lanczos approximation used here is accurate to about 15 significant digits, which is the precision limit of 64-bit floating-point arithmetic. Use this as a practical reminder before finalizing the result.
The gamma function appears in the probability density functions of the gamma, chi-squared, Student-t, F, beta, and Dirichlet distributions, and in the normalization constants of Bayesian models. Keep this note short and outcome-focused for reuse.