# Fungrim entry: 9a0bc8

$R_D\!\left(0, y, z\right) = 3 \int_{0}^{\pi / 2} \frac{\sin^{2}\!\left(\theta\right)}{{\left(y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)\right)}^{3 / 2}} \, d\theta$
Assumptions:$y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0$
TeX:
R_D\!\left(0, y, z\right) = 3 \int_{0}^{\pi / 2} \frac{\sin^{2}\!\left(\theta\right)}{{\left(y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)\right)}^{3 / 2}} \, d\theta

y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
Definitions:
Fungrim symbol Notation Short description
CarlsonRD$R_D\!\left(x, y, z\right)$ Degenerate Carlson symmetric elliptic integral of the third kind
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
Sin$\sin(z)$ Sine
Cos$\cos(z)$ Cosine
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("9a0bc8"),
Formula(Equal(CarlsonRD(0, y, z), Mul(3, Integral(Div(Pow(Sin(theta), 2), Pow(Add(Mul(y, Pow(Cos(theta), 2)), Mul(z, Pow(Sin(theta), 2))), Div(3, 2))), For(theta, 0, Div(Pi, 2)))))),
Variables(y, z),
Assumptions(And(Element(y, CC), Element(z, CC), Greater(Re(y), 0), Greater(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