Evaluate Γ(z) fast with reliable accuracy for physics tasks every day now. Choose gamma or log-gamma outputs, then download CSV and PDF summaries instantly.
| z | Γ(z) | Notes |
|---|---|---|
| 0.5 | √π ≈ 1.77245 | Half-integer appears in thermal distributions. |
| 1 | 1 | Normalization constant in many integrals. |
| 5 | 24 | Because Γ(5) = 4!. |
| -0.5 | -2√π ≈ -3.54491 | Negative non-integers use reflection safely. |
The gamma function extends the factorial continuously: Γ(z) = ∫₀^∞ t^{z-1} e^{-t} dt, for Re(z) > 0.
To compute values accurately, this calculator uses the Lanczos approximation for lnΓ(z), then evaluates Γ(z) = exp(lnΓ(z)) when the magnitude fits in floating-point.
For negative non-integers, it applies the reflection identity: Γ(z) = π / (sin(πz) Γ(1−z)). Poles occur at non-positive integers where Γ(z) is undefined.
Γ(z) appears in partition functions, power-law spectra, normalization of probability densities, and dimensional regularization integrals where factorial-like growth must be extended smoothly.
The gamma function extends factorial growth to non-integer arguments, which is essential in many physics formulas. It appears whenever integrals contain power laws multiplied by exponentials, a common structure in thermal, transport, and field theories. Typical inputs include half-integers from Gaussian integrals and dimension-dependent terms like d/2.
For positive integers, Γ(n) equals (n−1)!, so Γ(5)=24 and Γ(11)=10!=3,628,800. This relationship is used when counting microstates, arranging oscillator modes, or simplifying combinatorial prefactors in partition functions. The calculator helps verify identities and quickly compare Γ(z) against Stirling-style expectations for large z.
Half-integer values produce closed forms involving √π, such as Γ(1/2)=√π≈1.77245 and Γ(3/2)=√π/2≈0.88623. These constants show up in Maxwell–Boltzmann speed distributions, Gaussian normalization, and harmonic oscillator wavefunctions. They also appear in error-function derivations and diffusion solutions with quadratic exponents.
Γ(z) has poles at 0, −1, −2, …, so physical expressions must avoid those points. For negative non-integers, reflection relates Γ(z) to Γ(1−z) through sin(πz), which can flip the sign. For example, Γ(−1/2)=−2√π≈−3.54491, a value frequently encountered in analytic continuation steps.
Many problems produce extremely large Γ(z), especially when z represents a high-dimensional degree of freedom or a large shape parameter. Computing lnΓ(z) is numerically safer because it avoids overflow and preserves differences that matter in likelihood ratios, entropy terms, and free-energy comparisons. As a reference, Γ(171) is near floating-point limits, so lnΓ is preferred beyond that regime.
In d dimensions, surface areas and volumes of hyperspheres involve Γ(d/2). The unit-sphere volume is π^{d/2}/Γ(d/2+1), yielding 4π/3 in 3D and π^2/2 in 4D. These terms appear in density-of-states derivations, collision theory, and phase-space integrations where the measure depends on dimensionality.
Beta functions, factorial moments, and many Laplace-transform tables can be rewritten using gamma identities, improving clarity. Dimensional regularization often produces Γ(ε) and Γ(1−ε) factors, where the small-ε behavior controls divergences and renormalization constants. Gamma ratios also stabilize evaluations of combinatorial coefficients in perturbation series.
This tool uses a stable approximation for lnΓ(z) and applies reflection for small z to maintain accuracy. Choose scientific notation for extreme magnitudes, and export CSV or PDF to record inputs, outputs, sign information, and notes. Good reporting supports reproducibility in simulations, curve fitting, and classroom demonstrations involving special-function constants.
It evaluates the gamma function for real z and, if selected, lnΓ(z). The method combines a stable approximation with the reflection identity, which supports negative non-integers while warning about poles at 0, −1, −2, and so on.
Use lnΓ(z) when Γ(z) becomes extremely large or tiny. Log values avoid overflow and keep differences meaningful in likelihoods, free energies, and normalization factors. You can still export lnΓ(z) in CSV or PDF.
The gamma function has poles at non-positive integers. Near these points, sin(πz) approaches zero in the reflection relation, making the magnitude blow up. The calculator flags these regions because results become unstable or undefined.
Yes. For positive integers n, Γ(n) = (n−1)!. That means Γ(1)=1, Γ(5)=24, and Γ(11)=10!. This is useful for checking combinatorial factors in statistical mechanics derivations.
For typical real inputs, the Lanczos-based lnΓ(z) is highly accurate in double precision. Accuracy can degrade extremely close to poles or when cancellation occurs. Increasing displayed digits shows more detail, not necessarily more trustworthy digits.
For negative non-integers, the reflection identity introduces a sin(πz) factor that can be negative. That sign propagates into Γ(z). The calculator reports the sign separately so you can interpret results used inside physical prefactors.
Downloads contain the latest computed inputs and outputs stored during your session. The CSV is convenient for spreadsheets, while the PDF gives a one-page record for reports or lab notebooks, including any warning notes.
Accurate gamma estimates help interpret physics models more reliably.
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.