Fungrim entry: 944a14

R_D\!\left(x, y, z\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + x\right) \left(t + y\right) {\left(t + z\right)}^{3 / 2}} \, dt

Assumptions:

x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)

TeX:

R_D\!\left(x, y, z\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + x\right) \left(t + y\right) {\left(t + z\right)}^{3 / 2}} \, dt

x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRD`	$R_D\!\left(x, y, z\right)$	Degenerate Carlson symmetric elliptic integral of the third kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("944a14"),
    Formula(Equal(CarlsonRD(x, y, z), Mul(Div(3, 2), Integral(Div(1, Mul(Mul(Add(t, x), Add(t, y)), Pow(Add(t, z), Div(3, 2)))), For(t, 0, Infinity))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Or(NotEqual(x, 0), NotEqual(y, 0)))))

Topics using this entry

Carlson symmetric elliptic integrals