# Fungrim entry: e54e61

$R_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \begin{cases} 1 - i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 + i, & \text{otherwise}\\ \end{cases}$
Assumptions:$x \in \mathbb{C}$
TeX:
R_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \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
CarlsonRF$R_F\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the first kind
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
Gamma$\Gamma(z)$ Gamma function
Pi$\pi$ The constant pi (3.14...)
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("e54e61"),
Formula(Mul(Mul(Equal(CarlsonRF(0, x, Neg(x)), Div(1, Sqrt(x))), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi))))), Cases(Tuple(Sub(1, ConstI), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(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