Compute Γ(s), scaled integrals, and numerical estimates. Inspect curves, compare errors, and study parameter sensitivity. Designed for deeper analysis, learning, validation, and confident reporting.
The classic gamma function is defined for positive real s by the improper integral below.
This calculator also evaluates the scaled form with rate parameter β.
For additional interpretation, the graph’s peak occurs at x* = (s − 1)/β when s > 1. The calculator compares the exact formula with a numerical Simpson estimate over a practical upper limit.
| s | β | Expression | Exact Value | Mode x* |
|---|---|---|---|---|
| 0.5 | 1 | Γ(0.5) = √π | 1.772454 | 0.000000 |
| 1 | 1 | Γ(1) | 1.000000 | 0.000000 |
| 3 | 2 | Γ(3)/2³ | 0.250000 | 1.000000 |
| 5 | 1.5 | Γ(5)/1.5⁵ | 3.160494 | 2.666667 |
It evaluates the gamma integral and the scaled form with rate β. You get the exact formula value, a numerical approximation, error measures, a graph, and exportable summaries.
Yes. The gamma function is defined for positive real values, not only integers. Fractional inputs such as 0.5, 1.75, or 3.2 are valid and often useful.
β generalizes the classic integral. When β changes, the decay term changes too, and the integral scales as Γ(s)/βs. Setting β = 1 restores the standard gamma definition.
The numerical result uses Simpson’s rule over a finite upper limit, while the analytic formula represents the full improper integral. Larger limits and more intervals usually reduce the difference.
xmax is the truncation point used by the numerical estimate and graph. If it is too small, part of the tail is omitted, which increases the missing area and error.
When s is less than 1, the integrand becomes singular at x = 0. The graph therefore starts at a tiny positive value so the curve remains readable and stable.
For positive integers n, Γ(n) = (n − 1)!. That means Γ(5) = 4! = 24. This link makes the gamma function a continuous extension of factorials.
This page requires s > 0, β > 0, and a positive truncation limit. Nonpositive values break the intended formulas or make the numerical setup inappropriate.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.