Fungrim entry: c92da4

I_{x}\!\left(a, b\right) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\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) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\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
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	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("c92da4"),
    Formula(Equal(IncompleteBetaRegularized(x, a, b), Div(IncompleteBeta(x, a, b), BetaFunction(a, b)))),
    Variables(x, a, b),
    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