Specific values of Carlson symmetric elliptic integrals

a839d5

R_C\!\left(\lambda x, \lambda y\right) = {\lambda}^{-1 / 2} R_C\!\left(x, y\right)

Assumptions:

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

TeX:

R_C\!\left(\lambda x, \lambda y\right) = {\lambda}^{-1 / 2} R_C\!\left(x, y\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("a839d5"),
    Formula(Equal(CarlsonRC(Mul(lamda, x), Mul(lamda, y)), Mul(Pow(lamda, Neg(Div(1, 2))), CarlsonRC(x, y)))),
    Variables(x, y, lamda),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(lamda, OpenInterval(0, Infinity)))))

5c2b08

R_C\!\left(0, 0\right) = {\tilde \infty}

TeX:

R_C\!\left(0, 0\right) = {\tilde \infty}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("5c2b08"),
    Formula(Equal(CarlsonRC(0, 0), UnsignedInfinity)))

1acb07

R_C\!\left(1, 0\right) = \infty

TeX:

R_C\!\left(1, 0\right) = \infty

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("1acb07"),
    Formula(Equal(CarlsonRC(1, 0), Infinity)))

e464ec

R_C\!\left(0, 1\right) = \frac{\pi}{2}

TeX:

R_C\!\left(0, 1\right) = \frac{\pi}{2}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("e464ec"),
    Formula(Equal(CarlsonRC(0, 1), Div(Pi, 2))))

d38c27

R_C\!\left(1, 1\right) = 1

TeX:

R_C\!\left(1, 1\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind

Source code for this entry:

Entry(ID("d38c27"),
    Formula(Equal(CarlsonRC(1, 1), 1)))

eac389

R_C\!\left(1, 2\right) = \frac{\pi}{4}

TeX:

R_C\!\left(1, 2\right) = \frac{\pi}{4}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("eac389"),
    Formula(Equal(CarlsonRC(1, 2), Div(Pi, 4))))

a15c03

R_C\!\left(2, 1\right) = \log\!\left(1 + \sqrt{2}\right)

TeX:

R_C\!\left(2, 1\right) = \log\!\left(1 + \sqrt{2}\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Log`	$\log(z)$	Natural logarithm
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("a15c03"),
    Formula(Equal(CarlsonRC(2, 1), Log(Add(1, Sqrt(2))))))

35cb93

R_C\!\left(0, -1\right) = -\frac{\pi i}{2}

TeX:

R_C\!\left(0, -1\right) = -\frac{\pi i}{2}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("35cb93"),
    Formula(Equal(CarlsonRC(0, -1), Neg(Div(Mul(Pi, ConstI), 2)))))

56d1bc

R_C\!\left(-1, 0\right) = -i \infty

TeX:

R_C\!\left(-1, 0\right) = -i \infty

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("56d1bc"),
    Formula(Equal(CarlsonRC(-1, 0), Neg(Mul(ConstI, Infinity)))))

25435b

R_C\!\left(1, -1\right) = \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} - \frac{\pi \sqrt{2}}{4} i

TeX:

R_C\!\left(1, -1\right) = \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} - \frac{\pi \sqrt{2}}{4} i

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("25435b"),
    Formula(Equal(CarlsonRC(1, -1), Sub(Div(Mul(Sqrt(2), Log(Add(1, Sqrt(2)))), 2), Mul(Div(Mul(Pi, Sqrt(2)), 4), ConstI)))))

7ea1ad

R_C\!\left(-1, 1\right) = \frac{\pi \sqrt{2}}{4} - \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} i

TeX:

R_C\!\left(-1, 1\right) = \frac{\pi \sqrt{2}}{4} - \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} i

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`Log`	$\log(z)$	Natural logarithm
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("7ea1ad"),
    Formula(Equal(CarlsonRC(-1, 1), Sub(Div(Mul(Pi, Sqrt(2)), 4), Mul(Div(Mul(Sqrt(2), Log(Add(1, Sqrt(2)))), 2), ConstI)))))

7cbe17

R_C\!\left(x, 0\right) = \begin{cases} \operatorname{sgn}\!\left(\frac{1}{\sqrt{x}}\right) \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}

Assumptions:

x \in \mathbb{C}

TeX:

R_C\!\left(x, 0\right) = \begin{cases} \operatorname{sgn}\!\left(\frac{1}{\sqrt{x}}\right) \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sign`	$\operatorname{sgn}(z)$	Sign function
`Sqrt`	$\sqrt{z}$	Principal square root
`Infinity`	$\infty$	Positive infinity
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7cbe17"),
    Formula(Equal(CarlsonRC(x, 0), Cases(Tuple(Mul(Sign(Div(1, Sqrt(x))), Infinity), NotEqual(x, 0)), Tuple(UnsignedInfinity, Equal(x, 0))))),
    Variables(x),
    Assumptions(Element(x, CC)))

ff58cf

R_C\!\left(0, y\right) = \begin{cases} \frac{\pi}{2 \sqrt{y}}, & y \ne 0\\{\tilde \infty}, & y = 0\\ \end{cases}

Assumptions:

y \in \mathbb{C}

TeX:

R_C\!\left(0, y\right) = \begin{cases} \frac{\pi}{2 \sqrt{y}}, & y \ne 0\\{\tilde \infty}, & y = 0\\ \end{cases}

y \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ff58cf"),
    Formula(Equal(CarlsonRC(0, y), Cases(Tuple(Div(Pi, Mul(2, Sqrt(y))), NotEqual(y, 0)), Tuple(UnsignedInfinity, Equal(y, 0))))),
    Variables(y),
    Assumptions(Element(y, CC)))

ad96f4

R_C\!\left(x, x\right) = \frac{1}{\sqrt{x}}

Assumptions:

x \in \mathbb{C}

TeX:

R_C\!\left(x, x\right) = \frac{1}{\sqrt{x}}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ad96f4"),
    Formula(Equal(CarlsonRC(x, x), Div(1, Sqrt(x)))),
    Variables(x),
    Assumptions(Element(x, CC)))

09a494

R_C\!\left(x, 2 x\right) = \frac{\pi}{4 \sqrt{x}}

Assumptions:

x \in \mathbb{C}

TeX:

R_C\!\left(x, 2 x\right) = \frac{\pi}{4 \sqrt{x}}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("09a494"),
    Formula(Equal(CarlsonRC(x, Mul(2, x)), Div(Pi, Mul(4, Sqrt(x))))),
    Variables(x),
    Assumptions(Element(x, CC)))

b136bd

R_C\!\left(2 x, x\right) = \frac{\log\!\left(1 + \sqrt{2}\right)}{\sqrt{x}}

Assumptions:

x \in \mathbb{C}

TeX:

R_C\!\left(2 x, x\right) = \frac{\log\!\left(1 + \sqrt{2}\right)}{\sqrt{x}}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Log`	$\log(z)$	Natural logarithm
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b136bd"),
    Formula(Equal(CarlsonRC(Mul(2, x), x), Div(Log(Add(1, Sqrt(2))), Sqrt(x)))),
    Variables(x),
    Assumptions(Element(x, CC)))

5ada5f

R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}

Assumptions:

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

TeX:

R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Sqrt`	$\sqrt{z}$	Principal square root
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("5ada5f"),
    Formula(Equal(CarlsonRC(x, y), Cases(Tuple(Div(Atan(Sqrt(Sub(Div(y, x), 1))), Sqrt(Sub(y, x))), Less(x, y)), Tuple(Div(1, Sqrt(x)), Equal(x, y)), Tuple(Div(Atanh(Sqrt(Sub(1, Div(y, x)))), Sqrt(Sub(x, y))), Greater(x, y))))),
    Variables(x, y),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

718f3a

R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{acos}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{acosh}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}

Assumptions:

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

TeX:

R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{acos}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{acosh}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("718f3a"),
    Formula(Equal(CarlsonRC(x, y), Cases(Tuple(Div(Acos(Sqrt(Div(x, y))), Sqrt(Sub(y, x))), Less(x, y)), Tuple(Div(1, Sqrt(x)), Equal(x, y)), Tuple(Div(Acosh(Sqrt(Div(x, y))), Sqrt(Sub(x, y))), Greater(x, y))))),
    Variables(x, y),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

de0638

R_C\!\left(-x, -y\right) = -i R_C\!\left(x, y\right)

Assumptions:

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

TeX:

R_C\!\left(-x, -y\right) = -i R_C\!\left(x, y\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("de0638"),
    Formula(Equal(CarlsonRC(Neg(x), Neg(y)), Neg(Mul(ConstI, CarlsonRC(x, y))))),
    Variables(x, y),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

00cdb7

R_C\!\left(x, -y\right) = \frac{1}{\sqrt{x + y}} \left(\operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) - \frac{\pi i}{2}\right)

Assumptions:

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

TeX:

R_C\!\left(x, -y\right) = \frac{1}{\sqrt{x + y}} \left(\operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) - \frac{\pi i}{2}\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("00cdb7"),
    Formula(Equal(CarlsonRC(x, Neg(y)), Mul(Div(1, Sqrt(Add(x, y))), Sub(Atanh(Sqrt(Div(x, Add(x, y)))), Div(Mul(Pi, ConstI), 2))))),
    Variables(x, y),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

bc2f88

R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)

Assumptions:

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

TeX:

R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("bc2f88"),
    Formula(Equal(CarlsonRC(Neg(x), y), Mul(Div(1, Sqrt(Add(x, y))), Sub(Div(Pi, 2), Mul(Atanh(Sqrt(Div(x, Add(x, y)))), ConstI))))),
    Variables(x, y),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

4becdd

R_C\!\left(-x, y\right) = \overline{i R_C\!\left(x, -y\right)}

Assumptions:

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

TeX:

R_C\!\left(-x, y\right) = \overline{i R_C\!\left(x, -y\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Conjugate`	$\overline{z}$	Complex conjugate
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("4becdd"),
    Formula(Equal(CarlsonRC(Neg(x), y), Conjugate(Mul(ConstI, CarlsonRC(x, Neg(y)))))),
    Variables(x, y),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

7b5755

R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}

Assumptions:

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)

TeX:

R_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & 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)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("7b5755"),
    Formula(Equal(CarlsonRC(x, y), Cases(Tuple(Div(Atan(Sqrt(Sub(Div(y, x), 1))), Sqrt(Sub(y, x))), 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)))))))

0cf60d

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}

Assumptions:

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)

TeX:

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)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("0cf60d"),
    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)))))))

8c9ba1

R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \end{cases}

Assumptions:

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

TeX:

R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \end{cases}

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("8c9ba1"),
    Formula(Equal(CarlsonRC(x, Mul(c, x)), Cases(Tuple(Div(Atan(Sqrt(Sub(c, 1))), Sqrt(Mul(Sub(c, 1), x))), Greater(c, 1)), Tuple(Div(1, Sqrt(x)), Equal(c, 1)), Tuple(Div(Atanh(Sqrt(Sub(1, c))), Sqrt(Mul(Sub(1, c), x))), Less(c, 1))))),
    Variables(x, c),
    Assumptions(And(Element(x, CC), Element(c, OpenInterval(0, Infinity)))))

7348e3

R_C\!\left(x, -c x\right) = \frac{1}{\sqrt{\left(c + 1\right) x}} \begin{cases} \operatorname{atanh}\!\left(\sqrt{c + 1}\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\\operatorname{atanh}\!\left(\sqrt{c + 1}\right) + \pi i, & \text{otherwise}\\ \end{cases}

Assumptions:

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

TeX:

R_C\!\left(x, -c x\right) = \frac{1}{\sqrt{\left(c + 1\right) x}} \begin{cases} \operatorname{atanh}\!\left(\sqrt{c + 1}\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\\operatorname{atanh}\!\left(\sqrt{c + 1}\right) + \pi i, & \text{otherwise}\\ \end{cases}

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("7348e3"),
    Formula(Equal(CarlsonRC(x, Neg(Mul(c, x))), Mul(Div(1, Sqrt(Mul(Add(c, 1), x))), Cases(Tuple(Atanh(Sqrt(Add(c, 1))), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(Atanh(Sqrt(Add(c, 1))), Mul(Pi, ConstI)), Otherwise))))),
    Variables(x, c),
    Assumptions(And(Element(x, CC), Element(c, OpenInterval(0, Infinity)))))

eb1d4f

R_C\!\left(1, 1 + y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{y}\right)}{\sqrt{y}}, & y \ne 0\\1, & y = 0\\ \end{cases}

Assumptions:

y \in \mathbb{C}

TeX:

R_C\!\left(1, 1 + y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{y}\right)}{\sqrt{y}}, & y \ne 0\\1, & y = 0\\ \end{cases}

y \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("eb1d4f"),
    Formula(Equal(CarlsonRC(1, Add(1, y)), Cases(Tuple(Div(Atan(Sqrt(y)), Sqrt(y)), NotEqual(y, 0)), Tuple(1, Equal(y, 0))))),
    Variables(y),
    Assumptions(Element(y, CC)))

157ebb

R_C\!\left(1, 1 + y\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, -y\right)

Assumptions:

y \in \mathbb{C}

TeX:

R_C\!\left(1, 1 + y\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, -y\right)

y \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("157ebb"),
    Formula(Equal(CarlsonRC(1, Add(1, y)), Hypergeometric2F1(1, Div(1, 2), Div(3, 2), Neg(y)))),
    Variables(y),
    Assumptions(Element(y, CC)))

f29729

R_F\!\left(x, y, z\right) = R_F\!\left(x, z, y\right) = R_F\!\left(y, x, z\right) = R_F\!\left(y, z, x\right) = R_F\!\left(z, x, y\right) = R_F\!\left(z, y, x\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

R_F\!\left(x, y, z\right) = R_F\!\left(x, z, y\right) = R_F\!\left(y, x, z\right) = R_F\!\left(y, z, x\right) = R_F\!\left(z, x, y\right) = R_F\!\left(z, y, x\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f29729"),
    Formula(Equal(CarlsonRF(x, y, z), CarlsonRF(x, z, y), CarlsonRF(y, x, z), CarlsonRF(y, z, x), CarlsonRF(z, x, y), CarlsonRF(z, y, x))),
    Variables(x, y, z),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))

7a168a

R_F\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-1 / 2} R_F\!\left(x, y, z\right)

Assumptions:

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

TeX:

R_F\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-1 / 2} R_F\!\left(x, y, z\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("7a168a"),
    Formula(Equal(CarlsonRF(Mul(lamda, x), Mul(lamda, y), Mul(lamda, z)), Mul(Pow(lamda, Neg(Div(1, 2))), CarlsonRF(x, y, z)))),
    Variables(x, y, z, lamda),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(lamda, OpenInterval(0, Infinity)))))

e39456

R_F\!\left(0, 0, 0\right) = {\tilde \infty}

TeX:

R_F\!\left(0, 0, 0\right) = {\tilde \infty}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("e39456"),
    Formula(Equal(CarlsonRF(0, 0, 0), UnsignedInfinity)))

9a95a5

R_F\!\left(0, 0, 1\right) = \infty

TeX:

R_F\!\left(0, 0, 1\right) = \infty

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("9a95a5"),
    Formula(Equal(CarlsonRF(0, 0, 1), Infinity)))

8bb972

R_F\!\left(0, 1, 1\right) = \frac{\pi}{2}

TeX:

R_F\!\left(0, 1, 1\right) = \frac{\pi}{2}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("8bb972"),
    Formula(Equal(CarlsonRF(0, 1, 1), Div(Pi, 2))))

c166ca

R_F\!\left(1, 1, 1\right) = 1

TeX:

R_F\!\left(1, 1, 1\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind

Source code for this entry:

Entry(ID("c166ca"),
    Formula(Equal(CarlsonRF(1, 1, 1), 1)))

4cd504

R_F\!\left(1, 1, 2\right) = \log\!\left(1 + \sqrt{2}\right)

TeX:

R_F\!\left(1, 1, 2\right) = \log\!\left(1 + \sqrt{2}\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Log`	$\log(z)$	Natural logarithm
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("4cd504"),
    Formula(Equal(CarlsonRF(1, 1, 2), Log(Add(1, Sqrt(2))))))

0bf328

R_F\!\left(1, 2, 2\right) = \frac{\pi}{4}

TeX:

R_F\!\left(1, 2, 2\right) = \frac{\pi}{4}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("0bf328"),
    Formula(Equal(CarlsonRF(1, 2, 2), Div(Pi, 4))))

28237a

R_F\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}

TeX:

R_F\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("28237a"),
    Formula(Equal(CarlsonRF(0, 1, 2), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi)))))))

f1dd8a

R_F\!\left(0, 1, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right)

TeX:

R_F\!\left(0, 1, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("f1dd8a"),
    Formula(Equal(CarlsonRF(0, 1, -1), Mul(Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi)))), Sub(1, ConstI)))))

4c1988

R_F\!\left(0, 2, 4\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}}

TeX:

R_F\!\left(0, 2, 4\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("4c1988"),
    Formula(Equal(CarlsonRF(0, 2, 4), Div(Pow(Gamma(Div(1, 4)), 2), Mul(8, Sqrt(Pi))))))

6c4567

R_F\!\left(0, \frac{1}{2}, 1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}

TeX:

R_F\!\left(0, \frac{1}{2}, 1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("6c4567"),
    Formula(Equal(CarlsonRF(0, Div(1, 2), 1), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Pi))))))

90af98

R_F\!\left(0, 0, -1\right) = -i \infty

TeX:

R_F\!\left(0, 0, -1\right) = -i \infty

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("90af98"),
    Formula(Equal(CarlsonRF(0, 0, -1), Neg(Mul(ConstI, Infinity)))))

3a84d6

R_F\!\left(0, -1, -1\right) = \frac{-\pi i}{2}

TeX:

R_F\!\left(0, -1, -1\right) = \frac{-\pi i}{2}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("3a84d6"),
    Formula(Equal(CarlsonRF(0, -1, -1), Div(Neg(Mul(Pi, ConstI)), 2))))

6674bb

R_F\!\left(-1, -1, -1\right) = -i

TeX:

R_F\!\left(-1, -1, -1\right) = -i

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("6674bb"),
    Formula(Equal(CarlsonRF(-1, -1, -1), Neg(ConstI))))

5c178f

R_F\!\left(0, -1, -2\right) = -\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} i

TeX:

R_F\!\left(0, -1, -2\right) = -\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} i

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("5c178f"),
    Formula(Equal(CarlsonRF(0, -1, -2), Neg(Mul(Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi)))), ConstI)))))

e30d7e

R_F\!\left(0, 1, 12 \sqrt{2} - 16\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}}

TeX:

R_F\!\left(0, 1, 12 \sqrt{2} - 16\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("e30d7e"),
    Formula(Equal(CarlsonRF(0, 1, Sub(Mul(12, Sqrt(2)), 16)), Div(Mul(Add(2, Sqrt(2)), Pow(Gamma(Div(1, 4)), 2)), Mul(16, Sqrt(Pi))))))

cf5caa

R_F\!\left(0, i, -i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}

TeX:

R_F\!\left(0, i, -i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("cf5caa"),
    Formula(Equal(CarlsonRF(0, ConstI, Neg(ConstI)), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Pi))))))

67e015

R_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{16 \pi}, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1

TeX:

R_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{16 \pi}, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("67e015"),
    Formula(Equal(CarlsonRF(0, Div(Pow(Gamma(Div(1, 4)), 4), Mul(16, Pi)), Div(Pow(Gamma(Div(1, 4)), 4), Mul(32, Pi))), 1)))

8519dd

R_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}, \frac{-{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1 - i

TeX:

R_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}, \frac{-{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1 - i

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("8519dd"),
    Formula(Equal(CarlsonRF(0, Div(Pow(Gamma(Div(1, 4)), 4), Mul(32, Pi)), Div(Neg(Pow(Gamma(Div(1, 4)), 4)), Mul(32, Pi))), Sub(1, ConstI))))

13a092

R_F\!\left(0, 0, x\right) = \begin{cases} \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}

Assumptions:

x \in \mathbb{C}

TeX:

R_F\!\left(0, 0, x\right) = \begin{cases} \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Infinity`	$\infty$	Positive infinity
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("13a092"),
    Formula(Equal(CarlsonRF(0, 0, x), Cases(Tuple(Infinity, NotEqual(x, 0)), Tuple(UnsignedInfinity, Equal(x, 0))))),
    Variables(x),
    Assumptions(Element(x, CC)))

53d869

R_F\!\left(0, 1, x\right) = K\!\left(1 - x\right)

Assumptions:

x \in \mathbb{C}

TeX:

R_F\!\left(0, 1, x\right) = K\!\left(1 - x\right)

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("53d869"),
    Formula(Equal(CarlsonRF(0, 1, x), EllipticK(Sub(1, x)))),
    Variables(x),
    Assumptions(Element(x, CC)))

ab5af3

R_F\!\left(0, x, x\right) = \frac{\pi}{2 \sqrt{x}}

Assumptions:

x \in \mathbb{C}

TeX:

R_F\!\left(0, x, x\right) = \frac{\pi}{2 \sqrt{x}}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ab5af3"),
    Formula(Equal(CarlsonRF(0, x, x), Div(Pi, Mul(2, Sqrt(x))))),
    Variables(x),
    Assumptions(Element(x, CC)))

415ff0

R_F\!\left(0, x, y\right) = \frac{K\!\left(1 - \frac{y}{x}\right)}{\sqrt{x}}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(x) - \arg(y)\right| < \pi

TeX:

R_F\!\left(0, x, y\right) = \frac{K\!\left(1 - \frac{y}{x}\right)}{\sqrt{x}}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(x) - \arg(y)\right| < \pi

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg(z)$	Complex argument
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("415ff0"),
    Formula(Equal(CarlsonRF(0, x, y), Div(EllipticK(Sub(1, Div(y, x))), Sqrt(x)))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC), Less(Abs(Sub(Arg(x), Arg(y))), Pi))))

0ed5e2

R_F\!\left(0, x, 2 x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}

Assumptions:

x \in \mathbb{C}

TeX:

R_F\!\left(0, x, 2 x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0ed5e2"),
    Formula(Equal(CarlsonRF(0, x, Mul(2, x)), Mul(Div(1, Sqrt(x)), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi))))))),
    Variables(x),
    Assumptions(Element(x, CC)))

e54e61

R_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \begin{cases} 1 - i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 + i, & \text{otherwise}\\ \end{cases}

Assumptions:

x \in \mathbb{C}

TeX:

R_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \begin{cases} 1 - i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 + i, & \text{otherwise}\\ \end{cases}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e54e61"),
    Formula(Mul(Mul(Equal(CarlsonRF(0, x, Neg(x)), Div(1, Sqrt(x))), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi))))), Cases(Tuple(Sub(1, ConstI), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(1, ConstI), Otherwise)))),
    Variables(x),
    Assumptions(Element(x, CC)))

538c8c

R_F\!\left(0, x, c x\right) = \frac{K\!\left(1 - c\right)}{\sqrt{x}}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left[0, \infty\right)

TeX:

R_F\!\left(0, x, c x\right) = \frac{K\!\left(1 - c\right)}{\sqrt{x}}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("538c8c"),
    Formula(Equal(CarlsonRF(0, x, Mul(c, x)), Div(EllipticK(Sub(1, c)), Sqrt(x)))),
    Variables(x, c),
    Assumptions(And(Element(x, CC), Element(c, ClosedOpenInterval(0, Infinity)))))

271b73

R_F\!\left(0, x, -c x\right) = \frac{1}{\sqrt{x}} \begin{cases} K\!\left(1 + c\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\K\!\left(1 + c\right) + 2 i K\!\left(-c\right), & \text{otherwise}\\ \end{cases}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left[0, \infty\right)

TeX:

R_F\!\left(0, x, -c x\right) = \frac{1}{\sqrt{x}} \begin{cases} K\!\left(1 + c\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\K\!\left(1 + c\right) + 2 i K\!\left(-c\right), & \text{otherwise}\\ \end{cases}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("271b73"),
    Formula(Equal(CarlsonRF(0, x, Neg(Mul(c, x))), Mul(Div(1, Sqrt(x)), Cases(Tuple(EllipticK(Add(1, c)), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(EllipticK(Add(1, c)), Mul(Mul(2, ConstI), EllipticK(Neg(c)))), Otherwise))))),
    Variables(x, c),
    Assumptions(And(Element(x, CC), Element(c, ClosedOpenInterval(0, Infinity)))))

63d11e

R_F\!\left(x, y, y\right) = R_C\!\left(x, y\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}

TeX:

R_F\!\left(x, y, y\right) = R_C\!\left(x, y\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("63d11e"),
    Formula(Equal(CarlsonRF(x, y, y), CarlsonRC(x, y))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC))))

ebaa1a

R_F\!\left(x, x, y\right) = R_C\!\left(y, x\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}

TeX:

R_F\!\left(x, x, y\right) = R_C\!\left(y, x\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ebaa1a"),
    Formula(Equal(CarlsonRF(x, x, y), CarlsonRC(y, x))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC))))

649dc0

R_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(y \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \notin \left(-\infty, 0\right)\right)\right)

TeX:

R_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\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(y \in \left(0, \infty\right) \;\mathbin{\operatorname{or}}\; \left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \notin \left(-\infty, 0\right)\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("649dc0"),
    Formula(Equal(CarlsonRF(x, x, y), Cases(Tuple(Div(Atan(Sqrt(Sub(Div(x, y), 1))), 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(y, OpenInterval(0, Infinity)), And(Element(x, OpenInterval(0, Infinity)), NotElement(y, OpenInterval(Neg(Infinity), 0)))))))

9b0388

R_F\!\left(x, x, x\right) = \frac{1}{\sqrt{x}}

Assumptions:

x \in \mathbb{C}

TeX:

R_F\!\left(x, x, x\right) = \frac{1}{\sqrt{x}}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9b0388"),
    Formula(Equal(CarlsonRF(x, x, x), Div(1, Sqrt(x)))),
    Variables(x),
    Assumptions(Element(x, CC)))

5ab6bf

R_F\!\left(-x, -y, -z\right) = -i R_F\!\left(x, y, z\right)

Assumptions:

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

TeX:

R_F\!\left(-x, -y, -z\right) = -i R_F\!\left(x, y, z\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`ConstI`	$i$	Imaginary unit
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("5ab6bf"),
    Formula(Equal(CarlsonRF(Neg(x), Neg(y), Neg(z)), Neg(Mul(ConstI, CarlsonRF(x, y, z))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, ClosedOpenInterval(0, Infinity)), Element(z, ClosedOpenInterval(0, Infinity)))))

23e0a7

R_F\!\left(-x, -y, z\right) = \overline{i R_F\!\left(x, y, -z\right)}

Assumptions:

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

TeX:

R_F\!\left(-x, -y, z\right) = \overline{i R_F\!\left(x, y, -z\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Conjugate`	$\overline{z}$	Complex conjugate
`ConstI`	$i$	Imaginary unit
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("23e0a7"),
    Formula(Equal(CarlsonRF(Neg(x), Neg(y), z), Conjugate(Mul(ConstI, CarlsonRF(x, y, Neg(z)))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, ClosedOpenInterval(0, Infinity)), Element(z, ClosedOpenInterval(0, Infinity)))))

b478a1

R_G\!\left(x, y, z\right) = R_G\!\left(x, z, y\right) = R_G\!\left(y, x, z\right) = R_G\!\left(y, z, x\right) = R_G\!\left(z, x, y\right) = R_G\!\left(z, y, x\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

R_G\!\left(x, y, z\right) = R_G\!\left(x, z, y\right) = R_G\!\left(y, x, z\right) = R_G\!\left(y, z, x\right) = R_G\!\left(z, x, y\right) = R_G\!\left(z, y, x\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b478a1"),
    Formula(Equal(CarlsonRG(x, y, z), CarlsonRG(x, z, y), CarlsonRG(y, x, z), CarlsonRG(y, z, x), CarlsonRG(z, x, y), CarlsonRG(z, y, x))),
    Variables(x, y, z),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))

f9ca94

R_G\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{1 / 2} R_G\!\left(x, y, z\right)

Assumptions:

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

TeX:

R_G\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{1 / 2} R_G\!\left(x, y, z\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("f9ca94"),
    Formula(Equal(CarlsonRG(Mul(lamda, x), Mul(lamda, y), Mul(lamda, z)), Mul(Pow(lamda, Div(1, 2)), CarlsonRG(x, y, z)))),
    Variables(x, y, z, lamda),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(lamda, OpenInterval(0, Infinity)))))

bcc121

R_G\!\left(0, 0, 0\right) = 0

TeX:

R_G\!\left(0, 0, 0\right) = 0

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind

Source code for this entry:

Entry(ID("bcc121"),
    Formula(Equal(CarlsonRG(0, 0, 0), 0)))

d5ff09

R_G\!\left(0, 0, 1\right) = \frac{1}{2}

TeX:

R_G\!\left(0, 0, 1\right) = \frac{1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind

Source code for this entry:

Entry(ID("d5ff09"),
    Formula(Equal(CarlsonRG(0, 0, 1), Div(1, 2))))

cd55cf

R_G\!\left(0, 1, 1\right) = \frac{\pi}{4}

TeX:

R_G\!\left(0, 1, 1\right) = \frac{\pi}{4}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("cd55cf"),
    Formula(Equal(CarlsonRG(0, 1, 1), Div(Pi, 4))))

250ff1

R_G\!\left(1, 1, 1\right) = 1

TeX:

R_G\!\left(1, 1, 1\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind

Source code for this entry:

Entry(ID("250ff1"),
    Formula(Equal(CarlsonRG(1, 1, 1), 1)))

4d7098

R_G\!\left(1, 1, 2\right) = \frac{\sqrt{2}}{2} + \frac{\log\!\left(1 + \sqrt{2}\right)}{2}

TeX:

R_G\!\left(1, 1, 2\right) = \frac{\sqrt{2}}{2} + \frac{\log\!\left(1 + \sqrt{2}\right)}{2}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Log`	$\log(z)$	Natural logarithm

Source code for this entry:

Entry(ID("4d7098"),
    Formula(Equal(CarlsonRG(1, 1, 2), Add(Div(Sqrt(2), 2), Div(Log(Add(1, Sqrt(2))), 2)))))

d51efc

R_G\!\left(1, 2, 2\right) = \frac{\pi}{4} + \frac{1}{2}

TeX:

R_G\!\left(1, 2, 2\right) = \frac{\pi}{4} + \frac{1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("d51efc"),
    Formula(Equal(CarlsonRG(1, 2, 2), Add(Div(Pi, 4), Div(1, 2)))))

84f403

R_G\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

R_G\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("84f403"),
    Formula(Equal(CarlsonRG(0, 1, 2), Add(Div(Pow(Gamma(Div(1, 4)), 2), Mul(8, Sqrt(Mul(2, Pi)))), Div(Pow(Pi, Div(3, 2)), Mul(Sqrt(2), Pow(Gamma(Div(1, 4)), 2)))))))

9e30e7

R_G\!\left(0, 1, -1\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right)

TeX:

R_G\!\left(0, 1, -1\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("9e30e7"),
    Formula(Equal(CarlsonRG(0, 1, -1), Mul(Div(Mul(Sqrt(2), Pow(Pi, Div(3, 2))), Mul(2, Pow(Gamma(Div(1, 4)), 2))), Add(1, ConstI)))))

c5a9cf

R_G\!\left(0, 16, 16\right) = \pi

TeX:

R_G\!\left(0, 16, 16\right) = \pi

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("c5a9cf"),
    Formula(Equal(CarlsonRG(0, 16, 16), Pi)))

d829be

R_G\!\left(0, 0, x\right) = \frac{\sqrt{x}}{2}

Assumptions:

x \in \mathbb{C}

TeX:

R_G\!\left(0, 0, x\right) = \frac{\sqrt{x}}{2}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d829be"),
    Formula(Equal(CarlsonRG(0, 0, x), Div(Sqrt(x), 2))),
    Variables(x),
    Assumptions(Element(x, CC)))

3f6d40

R_G\!\left(0, 1, x\right) = \frac{E\!\left(1 - x\right)}{2}

Assumptions:

x \in \mathbb{C}

TeX:

R_G\!\left(0, 1, x\right) = \frac{E\!\left(1 - x\right)}{2}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("3f6d40"),
    Formula(Equal(CarlsonRG(0, 1, x), Div(EllipticE(Sub(1, x)), 2))),
    Variables(x),
    Assumptions(Element(x, CC)))

cdb587

R_G\!\left(0, x, x\right) = \frac{\pi \sqrt{x}}{4}

Assumptions:

x \in \mathbb{C}

TeX:

R_G\!\left(0, x, x\right) = \frac{\pi \sqrt{x}}{4}

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Pi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("cdb587"),
    Formula(Equal(CarlsonRG(0, x, x), Div(Mul(Pi, Sqrt(x)), 4))),
    Variables(x),
    Assumptions(Element(x, CC)))

7cddc6

R_G\!\left(0, x, y\right) = \frac{\sqrt{x} E\!\left(1 - \frac{y}{x}\right)}{2}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(x) - \arg(y)\right| < \pi

TeX:

R_G\!\left(0, x, y\right) = \frac{\sqrt{x} E\!\left(1 - \frac{y}{x}\right)}{2}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(x) - \arg(y)\right| < \pi

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg(z)$	Complex argument
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("7cddc6"),
    Formula(Equal(CarlsonRG(0, x, y), Div(Mul(Sqrt(x), EllipticE(Sub(1, Div(y, x)))), 2))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC), Less(Abs(Sub(Arg(x), Arg(y))), Pi))))

3f1547

R_G\!\left(0, x, 2 x\right) = \sqrt{x} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)

Assumptions:

x \in \mathbb{C}

TeX:

R_G\!\left(0, x, 2 x\right) = \sqrt{x} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("3f1547"),
    Formula(Equal(CarlsonRG(0, x, Mul(2, x)), Mul(Sqrt(x), Mul(Add(Div(Pow(Gamma(Div(1, 4)), 2), Mul(8, Sqrt(Mul(2, Pi)))), Div(Pow(Pi, Div(3, 2)), Mul(Sqrt(2), Pow(Gamma(Div(1, 4)), 2)))))))),
    Variables(x),
    Assumptions(Element(x, CC)))

7c50d1

R_G\!\left(0, x, -x\right) = \sqrt{x} \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \begin{cases} 1 + i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 - i, & \text{otherwise}\\ \end{cases}

The elementary integral RC

Scale invariance

Particular constant values

Specialized values

General formulas for real variables

General formulas for one or more complex variables

The integral of the first kind RF

Symmetry and scale invariance

Particular constant values

Special parametric cases

The integral of the second kind RG

Symmetry and scale invariance

Particular constant values

Special parametric cases