Fungrim entry: 8d0629

R_D\!\left(0, y, z\right) = {z}^{-3 / 2} \begin{cases} \frac{3 \left(K\!\left(1 - \frac{y}{z}\right) - E\!\left(1 - \frac{y}{z}\right)\right)}{1 - \frac{y}{z}}, & y \ne z\\\frac{3 \pi}{4}, & y = z\\ \end{cases}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(y) - \arg(z)\right| < \pi

TeX:

R_D\!\left(0, y, z\right) = {z}^{-3 / 2} \begin{cases} \frac{3 \left(K\!\left(1 - \frac{y}{z}\right) - E\!\left(1 - \frac{y}{z}\right)\right)}{1 - \frac{y}{z}}, & y \ne z\\\frac{3 \pi}{4}, & y = z\\ \end{cases}

z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(y) - \arg(z)\right| < \pi

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRD`	$R_D\!\left(x, y, z\right)$	Degenerate Carlson symmetric elliptic integral of the third kind
`Pow`	${a}^{b}$	Power
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg(z)$	Complex argument

Source code for this entry:

Entry(ID("8d0629"),
    Formula(Equal(CarlsonRD(0, y, z), Mul(Pow(z, Neg(Div(3, 2))), Cases(Tuple(Div(Mul(3, Sub(EllipticK(Sub(1, Div(y, z))), EllipticE(Sub(1, Div(y, z))))), Sub(1, Div(y, z))), NotEqual(y, z)), Tuple(Div(Mul(3, Pi), 4), Equal(y, z)))))),
    Variables(y, z),
    Assumptions(And(Element(z, SetMinus(CC, Set(0))), Element(y, CC), Less(Abs(Sub(Arg(y), Arg(z))), Pi))))

Fungrim entry: 8d0629

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