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Fungrim entry: 8d0629

RD ⁣(0,y,z)=z3/2{3(K ⁣(1yz)E ⁣(1yz))1yz,yz3π4,y=zR_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:zC{0}  and  yC  and  arg(y)arg(z)<πz \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(y) - \arg(z)\right| < \pi
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
Fungrim symbol Notation Short description
CarlsonRDRD ⁣(x,y,z)R_D\!\left(x, y, z\right) Degenerate Carlson symmetric elliptic integral of the third kind
Powab{a}^{b} Power
EllipticKK(m)K(m) Legendre complete elliptic integral of the first kind
EllipticEE(m)E(m) Legendre complete elliptic integral of the second kind
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Argarg(z)\arg(z) Complex argument
Source code for this entry:
    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))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC