# Fungrim entry: 1fc63b

$W_{k}\!\left(z\right) = \operatorname{L_1} - \operatorname{L_2} + \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\; \text{ where } \operatorname{L_1} = \log(z) + 2 \pi i k,\;\operatorname{L_2} = \log(\operatorname{L_1}),\;\sigma = \frac{1}{\operatorname{L_1}},\;\tau = \frac{\operatorname{L_2}}{\operatorname{L_1}}$
Assumptions:$k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left|\sigma\right| < \frac{1}{4} \;\mathbin{\operatorname{and}}\; \left|\tau\right| < \frac{1}{4} \;\mathbin{\operatorname{and}}\; \left(k \ne 0 \;\mathbin{\operatorname{or}}\; \left|z\right| > 1\right)\; \text{ where } \operatorname{L_1} = \log(z) + 2 \pi i k,\;\operatorname{L_2} = \log(\operatorname{L_1}),\;\sigma = \frac{1}{\operatorname{L_1}},\;\tau = \frac{\operatorname{L_2}}{\operatorname{L_1}}$
TeX:
W_{k}\!\left(z\right) = \operatorname{L_1} - \operatorname{L_2} + \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\; \text{ where } \operatorname{L_1} = \log(z) + 2 \pi i k,\;\operatorname{L_2} = \log(\operatorname{L_1}),\;\sigma = \frac{1}{\operatorname{L_1}},\;\tau = \frac{\operatorname{L_2}}{\operatorname{L_1}}

k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left|\sigma\right| < \frac{1}{4} \;\mathbin{\operatorname{and}}\; \left|\tau\right| < \frac{1}{4} \;\mathbin{\operatorname{and}}\; \left(k \ne 0 \;\mathbin{\operatorname{or}}\; \left|z\right| > 1\right)\; \text{ where } \operatorname{L_1} = \log(z) + 2 \pi i k,\;\operatorname{L_2} = \log(\operatorname{L_1}),\;\sigma = \frac{1}{\operatorname{L_1}},\;\tau = \frac{\operatorname{L_2}}{\operatorname{L_1}}
Definitions:
Fungrim symbol Notation Short description
LambertW$W\!\left(z\right)$ Lambert W-function
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
StirlingCycle$\left[{n \atop k}\right]$ Unsigned Stirling number of the first kind
Infinity$\infty$ Positive infinity
Log$\log(z)$ Natural logarithm
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
ZZ$\mathbb{Z}$ Integers
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("1fc63b"),
Formula(Equal(LambertW(z, k), Where(Add(Sub(L_1, L_2), Sum(Sum(Mul(Mul(Mul(Div(Pow(-1, n), Factorial(m)), StirlingCycle(Add(n, m), Add(n, 1))), Pow(sigma, n)), Pow(tau, m)), For(m, 1, Infinity)), For(n, 0, Infinity))), Equal(L_1, Add(Log(z), Mul(Mul(Mul(2, Pi), ConstI), k))), Equal(L_2, Log(L_1)), Equal(sigma, Div(1, L_1)), Equal(tau, Div(L_2, L_1))))),
Variables(k, z),
Assumptions(Where(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0))), Less(Abs(sigma), Div(1, 4)), Less(Abs(tau), Div(1, 4)), Or(NotEqual(k, 0), Greater(Abs(z), 1))), Equal(L_1, Add(Log(z), Mul(Mul(Mul(2, Pi), ConstI), k))), Equal(L_2, Log(L_1)), Equal(sigma, Div(1, L_1)), Equal(tau, Div(L_2, L_1)))))

## 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