# Fungrim entry: 7fbbe8

$R_G\!\left(0, y, z\right) = \frac{1}{2} \int_{0}^{\pi / 2} \sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)} \, d\theta$
Assumptions:$y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0$
TeX:
R_G\!\left(0, y, z\right) = \frac{1}{2} \int_{0}^{\pi / 2} \sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)} \, d\theta

y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0
Definitions:
Fungrim symbol Notation Short description
CarlsonRG$R_G\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the second kind
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
Cos$\cos(z)$ Cosine
Sin$\sin(z)$ Sine
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("7fbbe8"),
Formula(Equal(CarlsonRG(0, y, z), Mul(Div(1, 2), Integral(Sqrt(Add(Mul(y, Pow(Cos(theta), 2)), Mul(z, Pow(Sin(theta), 2)))), For(theta, 0, Div(Pi, 2)))))),
Variables(y, z),
Assumptions(And(Element(y, CC), Element(z, CC), GreaterEqual(Re(y), 0), GreaterEqual(Re(z), 0))))

## 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