Assumptions:
TeX:
R_G\!\left(0, x, -c x\right) = \frac{\sqrt{x}}{2} \begin{cases} E\!\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)\\E\!\left(1 + c\right) + 2 i \left(K\!\left(-c\right) - E\!\left(-c\right)\right), & \text{otherwise}\\ \end{cases}
x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \left[0, \infty\right)Definitions:
| Fungrim symbol | Notation | Short description |
|---|---|---|
| CarlsonRG | Carlson symmetric elliptic integral of the second kind | |
| Sqrt | Principal square root | |
| EllipticE | Legendre complete elliptic integral of the second kind | |
| Im | Imaginary part | |
| Re | Real part | |
| ConstI | Imaginary unit | |
| EllipticK | Legendre complete elliptic integral of the first kind | |
| CC | Complex numbers | |
| ClosedOpenInterval | Closed-open interval | |
| Infinity | Positive infinity |
Source code for this entry:
Entry(ID("48333c"),
Formula(Equal(CarlsonRG(0, x, Neg(Mul(c, x))), Mul(Div(Sqrt(x), 2), Cases(Tuple(EllipticE(Add(1, c)), Or(Less(Im(x), 0), And(Equal(Im(x), 0), GreaterEqual(Re(x), 0)))), Tuple(Add(EllipticE(Add(1, c)), Mul(Mul(2, ConstI), Sub(EllipticK(Neg(c)), EllipticE(Neg(c))))), Otherwise))))),
Variables(x, c),
Assumptions(And(Element(x, CC), Element(c, ClosedOpenInterval(0, Infinity)))))