Fungrim home page

Fungrim entry: dab889

RG ⁣(x,y,z)=140tt+xt+yt+z(xt+x+yt+y+zt+z)dtR_G\!\left(x, y, z\right) = \frac{1}{4} \int_{0}^{\infty} \frac{t}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) \, dt
Assumptions:xC(,0)  and  yC(,0)  and  zC(,0)x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right)
R_G\!\left(x, y, z\right) = \frac{1}{4} \int_{0}^{\infty} \frac{t}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) \, dt

x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right)
Fungrim symbol Notation Short description
CarlsonRGRG ⁣(x,y,z)R_G\!\left(x, y, z\right) Carlson symmetric elliptic integral of the second kind
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sqrtz\sqrt{z} Principal square root
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
OpenInterval(a,b)\left(a, b\right) Open interval
Source code for this entry:
    Formula(Equal(CarlsonRG(x, y, z), Mul(Div(1, 4), Integral(Mul(Div(t, Mul(Mul(Sqrt(Add(t, x)), Sqrt(Add(t, y))), Sqrt(Add(t, z)))), Add(Add(Div(x, Add(t, x)), Div(y, Add(t, y))), Div(z, Add(t, z)))), For(t, 0, Infinity))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenInterval(Neg(Infinity), 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