Fungrim entry: 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)))))

Topics using this entry

Specific values of Carlson symmetric elliptic integrals