# Fungrim entry: 7c50d1

$R_G\!\left(0, x, -x\right) = \sqrt{x} \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \begin{cases} 1 + i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 - i, & \text{otherwise}\\ \end{cases}$
Assumptions:$x \in \mathbb{C}$
TeX:
R_G\!\left(0, x, -x\right) = \sqrt{x} \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \begin{cases} 1 + i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 - i, & \text{otherwise}\\ \end{cases}

x \in \mathbb{C}
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
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Gamma$\Gamma(z)$ Gamma function
ConstI$i$ Imaginary unit
Im$\operatorname{Im}(z)$ Imaginary part
Re$\operatorname{Re}(z)$ Real part
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("7c50d1"),
Formula(Equal(CarlsonRG(0, x, Neg(x)), Mul(Mul(Sqrt(x), Div(Mul(Sqrt(2), Pow(Pi, Div(3, 2))), Mul(2, Pow(Gamma(Div(1, 4)), 2)))), Cases(Tuple(Add(1, ConstI), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Sub(1, ConstI), Otherwise))))),
Variables(x),
Assumptions(Element(x, CC)))

## 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