Fungrim entry: 315b3d

I_{x}\!\left(a, b\right) = 1 - I_{1 - x}\!\left(b, a\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

TeX:

I_{x}\!\left(a, b\right) = 1 - I_{1 - x}\!\left(b, a\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("315b3d"),
    Formula(Equal(IncompleteBetaRegularized(x, a, b), Sub(1, IncompleteBetaRegularized(Sub(1, x), b, a)))),
    Variables(a, b, x),
    Assumptions(And(Element(x, CC), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))), NotElement(Add(a, b), ZZLessEqual(0)))))

Topics using this entry

Beta function