# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC