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Fungrim entry: 99ff4c

Wk ⁣(z)=n=0μnvn   where v=2(ez+1)W_{k}\!\left(z\right) = \sum_{n=0}^{\infty} \mu_{n} {v}^{n}\; \text{ where } v = -\sqrt{2 \left(e z + 1\right)}
Assumptions:zCandez+1<1and((k=1andIm ⁣(z)0)or(k=1andIm ⁣(z)<0))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}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) < 0\right)\right)
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}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) < 0\right)\right)
Fungrim symbol Notation Short description
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
LambertWPuiseuxCoefficientμk\mu_{k} Coefficient in scaled Puiseux expansion of Lambert W-function
Powab{a}^{b} Power
Infinity\infty Positive infinity
Sqrtz\sqrt{z} Principal square root
ConstEee The constant e (2.718...)
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
Source code for this entry:
    Formula(Where(Equal(LambertW(k, z), Sum(Mul(LambertWPuiseuxCoefficient(n), Pow(v, n)), Tuple(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))))))

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

2019-09-15 11:00:55.020619 UTC