Fungrim entry: 398bb7

\log\!\left(\frac{x}{y}\right) = \left(x - y\right) R_C\!\left(\frac{{\left(x + y\right)}^{2}}{4}, x y\right)

Assumptions:

y \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \left(0, \infty\right)

TeX:

\log\!\left(\frac{x}{y}\right) = \left(x - y\right) R_C\!\left(\frac{{\left(x + y\right)}^{2}}{4}, x y\right)

y \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("398bb7"),
    Formula(Equal(Log(Div(x, y)), Mul(Sub(x, y), CarlsonRC(Div(Pow(Add(x, y), 2), 4), Mul(x, y))))),
    Variables(x, y),
    Assumptions(And(Element(y, OpenInterval(0, Infinity)), Element(x, OpenInterval(0, Infinity)))))

Topics using this entry

Carlson symmetric elliptic integrals