Log Gamma Function lnΓ(x) Calculator

Compute the log gamma function lnΓ(x) for any real x using the Lanczos approximation. It also reports the value and sign of Γ(x) and handles negative non-integers via the reflection formula.

Input

Enter a real number x to compute the log gamma function lnΓ(x) using the Lanczos approximation. The value and sign of Γ(x) are shown as well.

Argument x

Any real number. Zero and negative integers are poles and cannot be evaluated.

Result

lnΓ(4.5)

2.4537365708

Gamma function Γ(x)

11.6317284

Sign of Γ(x)

Positive (+)

Graph of lnΓ(x)

Showing ln|Γ(x)| over the interval [−4.5, 6]. The orange dot marks the current x.

How it works

lnΓ(x) is evaluated with the Lanczos approximation (g=7). Returning the logarithm of the factorial directly lets you handle large arguments without overflowing Γ(x).
For x below 0.5 the reflection formula Γ(x)Γ(1-x) = π / sin(π x) folds the argument before evaluation, so negative non-integers are supported.
Where Γ(x) is negative, lnΓ(x) is returned as ln|Γ(x)| with the sign reported separately. Zero and negative integers are poles of Γ and are undefined.

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