Gamma Function Calculator

The shift, and why it keeps producing wrong answers

x is a positive integer, so Γ(5) = 4! = 24. Note the shift: the argument is one MORE than the factorial being taken. This is the single most common mistake with the gamma function and it is a definition, not a convention anyone can change.

There is no way to argue this away. Legendre’s definition put the shift there, every table and library since has followed, and changing it now would break every reference in existence. The only defence is to state it, which this page does on every integer input.

The practical consequence: “the factorial of 4.5” means Γ(5.5), not Γ(4.5). Getting that wrong shifts the answer by a whole factor, and the result is usually plausible enough not to be caught.

Why Γ(½) is √π and not a convention

Γ(½) = 1.77245385091 = √π. That looks like a coincidence and is not — it is forced, and the forcing is worth seeing.

Substituting t = u² in the defining integral turns Γ(½) into 2∫₀∞ e⁻ᵘ² du, which is the Gaussian integral. Its value is √π, and so is Γ(½).

More fundamentally, the Bohr–Mollerup theorem says Γ is the ONLY function that interpolates the factorial while staying log-convex. Without that second condition there are infinitely many smooth interpolations; with it there is exactly one, and it has no freedom left to make Γ(½) anything else.

Between 1 and 2, where Γ has its only minimum for positive x: about 0.8856 at x ≈ 1.4616. It rises without bound on both sides of that.

The poles, and what the reflection formula explains

Γ has a simple pole at every non-positive integer, so Γ(-3) is undefined. The reflection formula shows why: Γ(x)Γ(1−x) = π/sin(πx), and sin(πx) is zero at every integer — so one of the two gammas has to be infinite there. For positive integers it is Γ(1−x) that blows up; for x ≤ 0 it is Γ(x) itself.

The reflection formula Γ(x)Γ(1−x) = π/sin(πx) accounts for the whole structure in one line. sin(πx) is zero at every integer, so the left side must be infinite there, so one of the two gammas blows up — and for x ≤ 0 it is Γ(x) itself.

It also produces the negative values. x is negative and not an integer, so Γ(x) is finite but sits between two poles at 0 and -1. Between each consecutive pair of poles Γ takes the opposite sign to the pair before, which is why its graph alternates below and above the axis as x goes more negative. Γ(−½) = −3.54490770181, which is −2√π, and the tool checks the reflection formula on it: Γ(−0.5) × Γ(1 − −0.5) = π / sin(π × −0.5) gives −3.141592654 against −3.141592654, which agree.

Between consecutive negative poles Γ alternates sign, which is why its graph flips above and below the axis as x runs more negative — another fact the reflection formula gives for free, since sin(πx) alternates sign between integers.

Past the overflow, and why Infinity is the wrong answer

Γ passes the largest double at about x = 171.6. Beyond that a naive implementation returns Infinity for numbers that exist perfectly well and have a few hundred digits.

Γ(175) is larger than the biggest double (about 1.8 × 10³⁰⁸), so the direct value overflows — Γ passes that ceiling at x ≈ 171.6. lnΓ(175) = 727.174567173 is an ordinary number, and the value above was reconstructed from it. Anything that would multiply or divide gammas this large should work in lnΓ throughout.

9.32096310408 × 10^156 for Γ(100.5) is comfortably inside the range; 6.425426 × 10^315 for Γ(175) is not, and comes from exponentiating lnΓ. The page marks which route each value took.

The general lesson for code: any expression that multiplies or divides large gammas should be written in lnΓ throughout and exponentiated once at the end, if at all. The beta function below is the standard example.

Digamma, and the derivative of a factorial

ψ(x) = Γ′(x)/Γ(x) is the logarithmic derivative — the rate at which lnΓ grows. It is the natural "derivative of the factorial", and like Γ itself it has no elementary closed form. ψ(1) = −γ = −0.5772156649, the negative of the Euler–Mascheroni constant, which is where that constant first appears in most courses.

ψ(x) = Γ′(x)/Γ(x) is the logarithmic derivative, and it is the natural answer to “what is the derivative of a factorial” — a question that has no elementary answer either, for the same reason Γ itself does not.

At x = 5 it is 1.506117668. It satisfies its own recurrence, ψ(x+1) = ψ(x) + 1/x, which makes ψ at integer arguments a harmonic number minus γ — the point where the Euler–Mascheroni constant enters most courses.

The beta function, and why it is computed in logarithms

B(a, b) = Γ(a)Γ(b)/Γ(a+b), and written that way it is unusable for large arguments: B(200, 300) = 1.64854916087 × 10^−147, while Γ(200), Γ(300) and Γ(500) are all past the overflow — the ratio as written is Infinity ÷ Infinity.

Computed as exp(lnΓ(a) + lnΓ(b) − lnΓ(a+b)) it is an ordinary small number. lnB = −337.980113065, and the value follows.

With whole-number arguments, 1/B(2, 3) = 4 × C(3, 1) = 12 — so the beta function is a binomial coefficient in disguise, which is exactly how it turns up in the beta distribution.

Which is why the beta function turns up wherever binomial coefficients do with non-integer arguments — the beta distribution, Bayesian priors on a probability, and the normalising constant of Student’s t.

Sources and methodology

The gamma function, its reflection formula and the beta function are standard; the references below carry the canonical statements.

• NIST Digital Library of Mathematical Functions — §5.2, Gamma and Psi Functions
• NIST DLMF — §5.5, Functional Relations
• How we calculate — methodology

Method. Every figure on this page comes from src/lib/gamma-function.ts over src/lib/algebra/special.ts. Γ uses the Lanczos approximation with g = 7 and nine coefficients, plus the reflection formula below x = ½; when the Lanczos product overflows on the way to a representable value, the result is recovered through lnΓ. The beta function is always computed in logarithms. That engine is verified on every change against 76 hand-written assertions, including agreement with exact big-integer factorials up to 171! to better than 1.6e-13 relative, and that the recurrence and reflection formulas hold at every tested argument. The count and the per-case breakdown are published on the formula verification page.

Related calculators

Where this goes next:

More in Math, or browse all calculators.

Read the guide

The gamma function is the parent of the incomplete gamma family, which is where the Error Function Calculator gets its precision. For exact factorials of whole numbers with no approximation at all, the Binomial Coefficient Calculator works on big integers instead.