Fungrim home page

Fungrim entry: 0cf60d

RC ⁣(x,y)={atanh ⁣(1yx)xy,xy1x,x=yR_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}
Assumptions:xC  and  yC  and  (x(0,)  or  (y(0,)  and  x(,0)))x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(y \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; x \notin \left(-\infty, 0\right)\right)\right)
R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(y \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; x \notin \left(-\infty, 0\right)\right)\right)
Fungrim symbol Notation Short description
CarlsonRCRC ⁣(x,y)R_C\!\left(x, y\right) Degenerate Carlson symmetric elliptic integral of the first kind
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Equal(CarlsonRC(x, y), Cases(Tuple(Div(Atanh(Sqrt(Sub(1, Div(y, x)))), Sqrt(Sub(x, y))), NotEqual(x, y)), Tuple(Div(1, Sqrt(x)), Equal(x, y))))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC), Or(Element(x, OpenInterval(0, Infinity)), And(Element(y, OpenInterval(0, Infinity)), NotElement(x, 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