Fungrim entry: a1941b

I_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

Assumptions:

x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

TeX:

I_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("a1941b"),
    Formula(Equal(IncompleteBetaRegularized(x, a, b), Mul(Div(1, BetaFunction(a, b)), Integral(Mul(Pow(t, Sub(a, 1)), Pow(Sub(1, t), Sub(b, 1))), For(t, 0, x))))),
    Variables(x, a, b),
    Assumptions(And(Element(x, ClosedInterval(0, 1)), Element(a, CC), Element(b, CC), Greater(Re(a), 0), Greater(Re(b), 0))))

Topics using this entry

Beta function