Fungrim entry: 99ff4c

W_{k}\!\left(z\right) = \sum_{n=0}^{\infty} \mu_{n} {v}^{n}\; \text{ where } v = -\sqrt{2 \left(e z + 1\right)}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|e z + 1\right| < 1 \;\mathbin{\operatorname{and}}\; \left(\left(k = -1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(k = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) < 0\right)\right)

TeX:

W_{k}\!\left(z\right) = \sum_{n=0}^{\infty} \mu_{n} {v}^{n}\; \text{ where } v = -\sqrt{2 \left(e z + 1\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|e z + 1\right| < 1 \;\mathbin{\operatorname{and}}\; \left(\left(k = -1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(k = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) < 0\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W\!\left(z\right)$	Lambert W-function
`Sum`	$\sum_{n} f(n)$	Sum
`LambertWPuiseuxCoefficient`	$\mu_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part

Source code for this entry:

Entry(ID("99ff4c"),
    Formula(Where(Equal(LambertW(z, k), Sum(Mul(LambertWPuiseuxCoefficient(n), Pow(v, n)), For(n, 0, Infinity))), Equal(v, Neg(Sqrt(Mul(2, Add(Mul(ConstE, z), 1))))))),
    Variables(k, z),
    Assumptions(And(Element(z, CC), Less(Abs(Add(Mul(ConstE, z), 1)), 1), Or(And(Equal(k, -1), GreaterEqual(Im(z), 0)), And(Equal(k, 1), Less(Im(z), 0))))))

Topics using this entry

Lambert W-function