# Fungrim entry: 48333c

$R_G\!\left(0, x, -c x\right) = \frac{\sqrt{x}}{2} \begin{cases} E\!\left(1 + c\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\E\!\left(1 + c\right) + 2 i \left(K\!\left(-c\right) - E\!\left(-c\right)\right), & \text{otherwise}\\ \end{cases}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left[0, \infty\right)$
TeX:
R_G\!\left(0, x, -c x\right) = \frac{\sqrt{x}}{2} \begin{cases} E\!\left(1 + c\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\E\!\left(1 + c\right) + 2 i \left(K\!\left(-c\right) - E\!\left(-c\right)\right), & \text{otherwise}\\ \end{cases}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left[0, \infty\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonRG$R_G\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the second kind
Sqrt$\sqrt{z}$ Principal square root
EllipticE$E(m)$ Legendre complete elliptic integral of the second kind
Im$\operatorname{Im}(z)$ Imaginary part
Re$\operatorname{Re}(z)$ Real part
ConstI$i$ Imaginary unit
EllipticK$K(m)$ Legendre complete elliptic integral of the first kind
CC$\mathbb{C}$ Complex numbers
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("48333c"),
Formula(Equal(CarlsonRG(0, x, Neg(Mul(c, x))), Mul(Div(Sqrt(x), 2), Cases(Tuple(EllipticE(Add(1, c)), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(EllipticE(Add(1, c)), Mul(Mul(2, ConstI), Sub(EllipticK(Neg(c)), EllipticE(Neg(c))))), Otherwise))))),
Variables(x, c),
Assumptions(And(Element(x, CC), Element(c, ClosedOpenInterval(0, Infinity)))))

## Topics using this entry

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