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Fungrim entry: 1fc63b

Wk ⁣(z)=L1L2+n=0m=1(1)nm![n+mn+1]σnτm   where L1=log(z)+2πik,  L2=log(L1),  σ=1L1,  τ=L2L1W_{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:kZ  and  zC{0}  and  σ<14  and  τ<14  and  (k0  or  z>1)   where L1=log(z)+2πik,  L2=log(L1),  σ=1L1,  τ=L2L1k \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
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Factorialn!n ! Factorial
StirlingCycle[nk]\left[{n \atop k}\right] Unsigned Stirling number of the first kind
Infinity\infty Positive infinity
Loglog(z)\log(z) Natural logarithm
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("1fc63b"),
    Formula(Equal(LambertW(k, z), 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)))))

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2020-04-08 16:14:44.404316 UTC