This gamma function calculator evaluates Γ(x) — the extension of the factorial to non-whole numbers — for any positive real input, so Γ(5) = 24 like 4!, and Γ(½) = √π ≈ 1.7725, with everything between and beyond.

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The Factorial, Continued

Factorials count arrangements of whole numbers of things, and they stop dead at fractions — there is no elementary sense in which "3.5!" means anything. Yet the factorial's growth pattern begs to be interpolated, and in 1729 Euler found the interpolating function: the gamma function, defined by the integral Γ(x) = ∫₀^∞ t^(x−1)e^(−t) dt and obeying the factorial's own recursion, Γ(x+1) = xΓ(x), with Γ(1) = 1. Immediately Γ(n) = (n−1)! at whole numbers: Γ(5) = 24, Γ(6) = 120. But the integral doesn't care about wholeness — Γ(½) = √π ≈ 1.7725, Γ(1.5) = √π/2 ≈ 0.8862, Γ(3.5) = 3.3234 — which lets you speak of "half-factorials" rigorously and, more importantly, powers the gamma, chi-square and t distributions of statistics, whose normalizing constants are gamma values. This calculator evaluates Γ for any positive real via the Lanczos approximation — the standard numerical method (g = 7, nine coefficients) that scientific libraries use — reaching about seven significant figures, from the gentle minimum near x ≈ 1.46 (Γ ≈ 0.8856, the function's only dip below 1 on the positive axis) to the vertical wall of large arguments where Γ(100) ≈ 9.33×10¹⁵⁵ overflows naive arithmetic and only the log-gamma form survives.

Why this particular interpolation is THE right one is a theorem, not a taste: Bohr and Mollerup proved (1922) that the gamma function is the UNIQUE log-convex function satisfying the factorial recursion with Γ(1) = 1 — impose the natural regularity condition and no other curve can impersonate the factorial between the integers. Around that uniqueness grew an empire: the gamma distribution in probability (waiting times for the x-th event), the factorial's appearances in series expansions and the Riemann zeta function's functional equation (Γ is the zeta's mirror partner), volumes of n-dimensional spheres (which involve Γ(n/2) — the reason π alternates with rational factors as dimension rises), and the Beta function B(x,y) = Γ(x)Γ(y)/Γ(x+y) tying it to the statistics of proportions. The reflection formula Γ(x)Γ(1−x) = π/sin(πx) — used inside this calculator's code to evaluate small arguments — is the same identity Euler prized. Small function, long shadow: enter any positive number and meet the factorial's true continuation.

Worked Examples (Verified)

  • Γ(½) = √π = 1.7724539 — the most famous gamma value.
  • Γ(5) = 4! = 24; Γ(6) = 5! = 120; Γ(1) = 0! = 1.
  • Γ(1.5) = ½√π = 0.8862269.
  • Γ(3.5) = 2.5 × 1.5 × 0.5 × √π = 3.3234.
  • Γ(10) = 9! = 362,880.

Gamma Function Discipline

  • Mind the shift: Γ(n) = (n−1)!, not n!. Half the gamma errors in the wild are this single index slip.
  • Poles at 0, −1, −2, …: Γ is undefined there (it dives to ±∞), which is why "the factorial of −1" has no answer — this calculator's positive-domain restriction is mathematics, not timidity.
  • For x above ~171, Γ(x) exceeds double-precision range (~1.8×10³⁰⁸): use ln Γ (the lgamma function) and keep results in log space.
  • Half-integer values all reduce to √π times a rational: Γ(n + ½) = (2n)!/(4ⁿn!)·√π — a closed form worth using to check the numerics.
  • Γ is real and smooth for positive x but oscillates wildly in sign for negative non-integers — same recurrence, very different neighborhood.

Related Tools

Frequently Asked Questions

How do you calculate the gamma function?

Γ(x) = ∫₀^∞ t^(x−1) e^(−t) dt — evaluated numerically in practice, since the integral has no closed form for general x. This calculator uses the Lanczos approximation (the algorithm in most scientific software) to about 7 significant figures, with the reflection formula for small arguments. At whole numbers no machinery is needed: Γ(n) = (n−1)!. At half-integers, Γ(n + ½) reduces to √π times a rational number — Γ(½) = 1.7725 is the anchor of the whole family.

Why is Γ(½) equal to √π?

Substitute t = u² in the integral and Γ(½) becomes twice the Gaussian integral ∫₀^∞ e^(−u²) du = √π — the most famous integral in probability. So the factorial of −½ (since Γ(½) = (−½)! in the shifted notation) is √π, and π enters every normal-distribution density through exactly this back door: the 1/√(2π) in the bell curve is gamma arithmetic. It's also why half-integer gamma values always carry √π as a factor.

What is the difference between the gamma function and factorial?

The factorial is defined only at non-negative integers (with 0! = 1); the gamma function is its unique smooth, log-convex continuation to all positive reals (and most negative ones), satisfying Γ(n) = (n−1)! at the integers. Note the shift by one — a historical quirk (Legendre's normalization) that forever trips up newcomers: think "gamma of n is the factorial of the number BELOW n." Everything the factorial does for counting, gamma does for the continuous distributions built on it.

Where is the gamma function used?

Everywhere continuous factorials hide: the gamma, chi-square, F and t distributions of statistics (their density constants are gamma values); the Beta function and Dirichlet distributions in Bayesian work; volumes and surface areas of high-dimensional spheres in physics and machine learning; the Riemann zeta function's functional equation in number theory; and Stirling's approximation's exact form, Γ(x) ≈ √(2π/x)(x/e)^x, which this function's log satisfies. If a formula needs "(something)!" of a non-integer, it is calling gamma.

What is the value of Γ at its minimum?

On the positive real axis, Γ dips once: the minimum sits at x ≈ 1.4616, where Γ ≈ 0.8856 — the only place the positive-domain function falls below 1. The defaults' neighborhood illustrates it: Γ(1) = 1, Γ(1.5) = 0.8862 (nearly the minimum), Γ(2) = 1 again, then factorial growth takes over irreversibly. The dip matters practically in the gamma distribution's shape parameter region and is a handy numeric check: any gamma implementation that doesn't dip below 1 between 1 and 2 is broken.

Why can't I compute the gamma of negative integers?

Because the function has poles there — it blows up to infinity as x approaches 0, −1, −2, … from either side. The recursion Γ(x) = Γ(x+1)/x shows the mechanism: reaching Γ(0) demands dividing Γ(1) by 0. The integral definition likewise diverges. Between the poles, negative non-integer values exist and alternate in sign (Γ(−0.5) = −2√π ≈ −3.545), but the negative integers themselves are genuine singularities, not gaps in the calculator — mathematics' own "no answer."

How accurate is this gamma function calculator?

About 7 significant figures across the positive domain, from the Lanczos approximation with the standard g = 7 coefficient set — the same accuracy class as the gamma functions in Excel, Python's math.gamma and most CAS tools for everyday work. Check it against the closed forms: Γ(½) = 1.7724539 (√π), Γ(5) = 24 exactly, Γ(3.5) = 3.3234. For arguments above ~171 the value itself exceeds floating-point range and only ln Γ stays meaningful — the calculator reports in scientific notation up to that wall.

Is this gamma function calculator free?

Yes — free, instant, no sign-up. Enter any positive real number for Γ(x), from half-integers like Γ(½) = √π to full factorials like Γ(10) = 9!.