Beta function

Beta functions for real and complex values

Defined via the analytic continuation of the Gamma function.

boring_math.special_functions.gamma_family.beta.beta(u: complex, v: complex) → complex

Beta function for all complex u and v.

param u:

First argument to analytically continued beta function.

param v:

Second argument to analytically continued beta function.

returns:

Value of beta(u,v) where inf+infj is returned to represent a single complex infinity.

Note

For all u, v ∈ ℂ and m, n ∈ ℕ, we have

• B(u, v) = B(v, u)

• B(0, v) = Γ(0) = ∞

• B(m, -n) = ∞ when m > n > 0

• B(m, -n) = Γ(m) * (Res[Γ, -n]/Res[Γ, m-n]) when m <= n

• where ∀(n>=0) Res[Γ(z), z = -n] = (-1)ⁿ/(n!)

Note

Using natural logs in calculation for more numerical stability.

boring_math.special_functions.gamma_family.beta.beta_real(x: float, y: float) → float

Beta function valid for all real values x, y > 1.

param x:

First argument to analytically continued beta function.

param y:

Second argument to analytically continued beta function.

returns:

Value of beta(x, y) where inf is returned to denote singular points.

raises ValueError:

If x <= 0 or y <= 0.

Note

Using natural logs for more numerical stability.

Warning

Not valid for extended value reals.