Beta function

Beta function.

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

Beta function valid for all complex values of z.

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.

Parameters:

• u – First argument to analytically continued beta function.

• 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.

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

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

Note

Not valid for extended value reals.

Note

Using natural logs for more numerical stability.

Parameters:

• x – First argument to analytically continued beta function.

• 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.