Incomplete Gamma Function Calculator

Compute the lower γ(s,x) and upper Γ(s,x) incomplete gamma plus regularized P(s,x) and Q(s,x) using series and continued fractions.

Input

Enter the parameters s and x to compute the lower and upper incomplete gamma functions and the regularized values P and Q.

Shape parameter s

Enter a positive value

Upper limit x

Enter a value of 0 or more

Lower incomplete gamma γ(s,x)

0.6466471676

s = 3, x = 2

Upper Γ(s,x)

1.3533528324

Regularized P(s,x)

0.3233235838

Regularized Q(s,x)

0.6766764162

Complete gamma Γ(s)

Chart of regularized P(s,t)

Value of P(s,t) against t on the horizontal axis. The orange dot marks the entered x.

The lower incomplete gamma γ(s,x) integrates t^(s-1)e^(-t) from 0 to x, while the upper Γ(s,x) integrates from x to infinity. Their sum equals the complete gamma Γ(s).
The regularized values P(s,x) = γ(s,x)/Γ(s) and Q(s,x) = Γ(s,x)/Γ(s) lie between 0 and 1 and satisfy P + Q = 1.
For x below s+1 the tool evaluates P(s,x) by series expansion, and for larger x it evaluates Q(s,x) by continued fraction, so the two methods cover complementary regions with high accuracy.
The complete gamma Γ(s) is obtained from ln(Γ(s)) via the Lanczos approximation. Use a positive s and a non-negative x.

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