Reference · Math

math.lgamma

math.lgamma(num) -> float

What it does

The natural logarithm of the absolute value of the gamma function, as Python's math.lgamma: math.lgamma(x) is log(|gamma(x)|), computed without forming gamma(x) itself, so it works far past where math.gamma overflows. math.lgamma(n + 1) is log(n!), the size of a factorial that no int or float could hold.

Examples

print(math.lgamma(5), math.log(24))print(math.lgamma(1), math.lgamma(2), math.lgamma(0.5))print(math.lgamma(1000))

3.1780538303479458 3.17805383034794580.0 0.0 0.57236494292470015905.220423209181

Edge cases

• math.lgamma(1) and math.lgamma(2) are 0.0, since 0! and 1! are 1; elsewhere it agrees with math.log(math.gamma(x)) to a float's last digit or so.
• The sign of gamma is dropped: math.lgamma(-0.5) is log(3.545...), positive, though math.gamma(-0.5) is negative.
• math.gamma(172) overflows a float but math.lgamma(172) is about 711.7; the number of digits of n! is math.lgamma(n + 1) / math.log(10), rounded up.
• Infinity gives infinity, not an error; NaN stays NaN.

print(math.lgamma(-0.5), math.log(-math.gamma(-0.5)))print(math.lgamma(172), math.ceil(math.lgamma(101) / math.log(10)))print(math.lgamma(float("inf")), math.lgamma(float("nan")))

1.2655121234846454 1.2655121234846454711.71472580229 158inf NaN

Errors

Zero and the negative ints are the poles of gamma, and a runtime error here too, math domain error, not an error value and not infinity, as Python's ValueError. Minus infinity is the same. An argument so large that even the logarithm overflows a float, math.lgamma(1e308), is a runtime error, math range error. An argument that is not a number is a runtime error as well.

print(math.lgamma(0))

line 1: `math.lgamma`: math domain error (0)

print(math.lgamma(-1))

line 1: `math.lgamma`: math domain error (-1)

Chaining

In a pipeline the number comes first. log(n choose k) without any overflow:

print(5 | math.lgamma)let log_comb = (n, k) => math.lgamma(n + 1) - math.lgamma(k + 1) - math.lgamma(n - k + 1)print(math.exp(log_comb(10, 3)) | round, math.comb(10, 3))

3.1780538303479458120 120

See also

math.gamma, math.factorial, math.log, math.comb

Edit this page on GitHub · every example's output was made by running it (tools/refcheck.mjs) · questions on Discord or r/quillo