Fungrim entry: e50532

W\!\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

TeX:

W\!\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

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

Source code for this entry:

Entry(ID("e50532"),
    Formula(Where(Equal(LambertW(z), Sum(Mul(LambertWPuiseuxCoefficient(n), Pow(v, n)), For(n, 0, Infinity))), Equal(v, Sqrt(Mul(2, Add(Mul(ConstE, z), 1)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(Add(Mul(ConstE, z), 1)), 1))))

Topics using this entry

Lambert W-function