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Function std.mathspecial.gammaIncompleteCompl

Regularized upper incomplete gamma function Q(a,x)

real gammaIncompleteCompl(
  real a,
  real x
) pure nothrow @nogc @safe;

Mathematically, Q(a,x) = Γ(a,x)/Γ(a), where Γ(a,x) is the upper incomplete gamma function, Γ(a,x) = ∫x∞ta-1e-tdt, a > 0 , and x ≥ 0. Note that P(a,x) + Q(a,x) = 1 or Q(a,x) = 1 - P(a,x), so Q is the complement of P.

Parameters

NameDescription
a the shape parameter, must be positive
x the fraction of integration completion from above, must be non-negative

Returns

It returns Q(a,x), an element of [0,1].

Special Values
a x gammaIncompleteCompl(a, x)
negative NAN
< 0 NAN
positive 0 1
positive ∞ 0
+0 > 0 0
∞ (0, ∞) 1

See Also

gamma and gammaIncomplete

Example

assert(isClose(gammaIncompleteCompl(2, 1), 2/E));
writeln(gammaIncompleteCompl(1, 0)); // 1
writeln(gammaIncompleteCompl(1, real.infinity)); // 0
writeln(gammaIncompleteCompl(+0., 1)); // 0
writeln(gammaIncompleteCompl(real.infinity, 1)); // 1
assert(isClose(gammaIncompleteCompl(1, 2), 1-gammaIncomplete(1, 2)));

Authors

Stephen L. Moshier (original C code). Conversion to D by Don Clugston

License

Boost License 1.0.