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Function `std.mathspecial.beta`

Beta function, B(x,y)


						
				real beta(
				
  real x,
				
  real y
				

				) pure nothrow @nogc @safe;

Mathematically, if x > 0 and y > 0 then B(x,y) = ∫₀¹t^x-1(l-t)^y-1dt. Through analytic continuation, it is extended to ℂ 2 where it can be expressed in terms of Γ(z).

B(x,y) = Γ(x)Γ(y) / Γ(x+y).

This implementation restricts x and y to the set of real numbers.

Parameters

Name	Description
x	the first argument of B
y	the second argument of B

Returns

It returns B(x,y) if it can be computed, otherwise NAN.

Special Values
x	y	beta(x, y)
NAN	y	NAN
-∞	y	NAN
integer < 0	y	NAN
noninteger and x+y even ≤ 0	noninteger	-0
noninteger and x+y odd ≤ 0	noninteger	+0
+0	positive finite	+∞
+0	+∞	NAN
> 0	+∞	+0
-0	+0	NAN
-0	> 0	-∞
noninteger < 0, ⌈x⌉ odd	+∞	-∞
noninteger < 0, ⌈x⌉ even	+∞	+∞
noninteger < 0	±0	±∞

Since B(x,y) = B(y,x), if the table states that beta(x, y) is a special value, then beta(y, x) is one as well.

Example

writeln(beta(1, 2)); // 0.5

Authors

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

License

Boost License 1.0.