Mathematics · Calculus · Integral Calculus
Gamma Function Calculator
Computes the Gamma function Γ(z) for real positive inputs using the integral definition and Lanczos approximation.
Calculator
Formula
Γ(z) is the Gamma function evaluated at z. The integral extends from 0 to ∞, where t is the integration variable, z is the input argument (must be a positive real number for this definition), and e is Euler's number. For positive integers n, Γ(n) = (n−1)!, generalizing the factorial to all positive reals. The key recurrence relation is Γ(z+1) = z·Γ(z), with the anchor value Γ(1) = 1.
Source: Abramowitz & Stegun, Handbook of Mathematical Functions, Chapter 6 (1964); DLMF §5.2
How it works
The Gamma function is defined by the improper integral Γ(z) = ∫₀^∞ t^(z−1) e^(−t) dt for all z with positive real part. This integral converges for z > 0 and extends the factorial in the sense that Γ(n) = (n−1)! for every positive integer n. So Γ(5) = 4! = 24, Γ(1) = 1, and Γ(1/2) = √π ≈ 1.7725. The fundamental recurrence Γ(z+1) = z·Γ(z) allows the function to be extended analytically to all complex numbers except the non-positive integers, where it has simple poles.
Because the integral definition is computationally impractical for general use, this calculator uses the Lanczos approximation, a highly efficient rational-function method accurate to machine precision for positive real arguments. The approximation expresses Γ(z) as a product of a power term and an exponential term scaled by a carefully chosen polynomial, requiring only a handful of floating-point operations. For z between 0 and 0.5, Euler's reflection formula Γ(z)·Γ(1−z) = π/sin(πz) is applied first to ensure numerical stability. The calculator also outputs ln Γ(z), which is preferred when Gamma values are astronomically large, as is common in Bayesian statistics and combinatorics.
Practical applications are extensive. In probability theory, the Gamma distribution's PDF involves Γ(α) as a normalizing constant. The Beta function B(a,b) = Γ(a)Γ(b)/Γ(a+b) underlies Bayesian conjugate priors. In physics, Gamma functions arise in quantum mechanics, the Riemann zeta function, and Feynman path integrals. Engineers use Γ(z) to compute generalized binomial coefficients for non-integer orders of differentiation in fractional calculus.
Worked example
Example 1 — Integer input (checking against factorial):
Let z = 5. We expect Γ(5) = 4! = 24.
Applying the Lanczos approximation: Γ(5) = 24.00000000. ✓
Example 2 — Half-integer input:
Let z = 0.5. The known exact value is Γ(1/2) = √π.
√π ≈ 1.77245385.
Calculator output: Γ(0.5) = 1.77245385. ✓
Example 3 — Non-integer real input:
Let z = 4.5. Using the recurrence: Γ(4.5) = 3.5 · Γ(3.5) = 3.5 · 2.5 · Γ(2.5) = 3.5 · 2.5 · 1.5 · Γ(1.5) = 3.5 · 2.5 · 1.5 · 0.5 · Γ(0.5) = 3.5 · 2.5 · 1.5 · 0.5 · √π ≈ 6.5625 · 1.77245385 ≈ 11.63172840.
Calculator output: Γ(4.5) = 11.63172840. ✓
Example 4 — Large input (using ln Γ):
Let z = 100. Γ(100) = 99! ≈ 9.333 × 10^155, which overflows many representations. The log-gamma output is ln Γ(100) ≈ 359.13434, which remains tractable. This illustrates why ln Γ(z) is the preferred output in statistical computing.
Limitations & notes
This calculator is designed for positive real inputs (z > 0). The Gamma function is undefined at zero and all negative integers due to simple poles; inputs at or near these values will return NaN or very large numbers. While the Lanczos approximation is highly accurate for moderate positive real arguments, extreme inputs (z > 170 for direct Gamma, very small z close to 0) may encounter floating-point overflow or underflow — use the ln Γ(z) output in these cases. The calculator does not support complex-valued inputs. For negative non-integer inputs, the reflection formula could extend evaluation, but this is not implemented here. Additionally, the recurrence Γ(z+1) = z·Γ(z) means that errors in small z values can propagate — the Lanczos implementation mitigates this by always reducing to a convenient range before computing. Results should be verified against tabulated values for mission-critical applications.
Frequently asked questions
What is the Gamma function used for in statistics?
In statistics, the Gamma function appears as a normalizing constant in several important probability distributions, including the Gamma, chi-squared, and Beta distributions. It is also fundamental to computing the multivariate Beta function used in Dirichlet distributions and Bayesian inference.
What is the relationship between the Gamma function and the factorial?
For any positive integer n, Γ(n) = (n−1)!. So Γ(1) = 1, Γ(2) = 1, Γ(3) = 2, Γ(4) = 6, and so on. The Gamma function extends this concept to all positive reals — and even complex numbers — not just the integers.
What is Γ(1/2) and why is it notable?
Γ(1/2) = √π ≈ 1.77245385. This is one of the most famous special values of the Gamma function, and it arises because the defining integral Γ(1/2) = ∫₀^∞ t^(−1/2) e^(−t) dt equals the Gaussian integral evaluated over the positive half-line. It appears frequently in probability theory, particularly in the chi-squared distribution.
Why is ln Γ(z) often preferred over Γ(z) itself?
For large values of z, Γ(z) grows faster than exponentially — for example, Γ(100) ≈ 9.3 × 10^155, which exceeds the range of standard 64-bit floating-point numbers. The natural logarithm ln Γ(z) grows much more slowly and remains numerically tractable. In statistical computing and Bayesian methods, log-probabilities are almost always used to avoid underflow and overflow.
What is the Lanczos approximation?
The Lanczos approximation, introduced by Cornelius Lanczos in 1964, is a method for computing the Gamma function to arbitrary precision using a finite sum of rational functions. It expresses Γ(z+1) as a product of powers and exponentials scaled by a polynomial sum with carefully tuned coefficients. With g = 7 and 9 coefficients, as used here, the approximation achieves machine double-precision accuracy (around 15 significant digits) for positive real arguments.
Last updated: 2025-01-15 · Formula verified against primary sources.