Fungrim entry: da0f15

\left|W_{k}\!\left(z\right) - \left({L}_{1} - {L}_{2} + \sum_{n=0}^{N - 1} \sum_{m=1}^{M - 1} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\right)\right| \le \frac{4 \left|\tau\right| {\left(4 \left|\sigma\right|\right)}^{N} + {\left(4 \left|\tau\right|\right)}^{M}}{\left(1 - 4 \left|\sigma\right|\right) \left(1 - 4 \left|\tau\right|\right)}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

Assumptions:

N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \left|\sigma\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left|\tau\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left(k \ne 0 \,\mathbin{\operatorname{or}}\, \left|z\right| \gt 1\right)\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

TeX:

\left|W_{k}\!\left(z\right) - \left({L}_{1} - {L}_{2} + \sum_{n=0}^{N - 1} \sum_{m=1}^{M - 1} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\right)\right| \le \frac{4 \left|\tau\right| {\left(4 \left|\sigma\right|\right)}^{N} + {\left(4 \left|\tau\right|\right)}^{M}}{\left(1 - 4 \left|\sigma\right|\right) \left(1 - 4 \left|\tau\right|\right)}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \left|\sigma\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left|\tau\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left(k \ne 0 \,\mathbin{\operatorname{or}}\, \left|z\right| \gt 1\right)\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("da0f15"),
    Formula(Where(LessEqual(Abs(Sub(LambertW(k, z), Parentheses(Add(Sub(Subscript(L, 1), Subscript(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)), Tuple(m, 1, Sub(M, 1))), Tuple(n, 0, Sub(N, 1))))))), Div(Add(Mul(Mul(4, Abs(tau)), Pow(Mul(4, Abs(sigma)), N)), Pow(Mul(4, Abs(tau)), M)), Mul(Sub(1, Mul(4, Abs(sigma))), Sub(1, Mul(4, Abs(tau)))))), Equal(Subscript(L, 1), Add(Log(z), Mul(Mul(Mul(2, ConstPi), ConstI), k))), Equal(Subscript(L, 2), Log(Subscript(L, 1))), Equal(sigma, Div(1, Subscript(L, 1))), Equal(tau, Div(Subscript(L, 2), Subscript(L, 1))))),
    Variables(k, z, N, M),
    Assumptions(Where(And(Element(N, ZZGreaterEqual(0)), Element(M, ZZGreaterEqual(0)), Element(k, ZZ), Element(z, SetMinus(CC, Set(0))), Less(Abs(sigma), Div(1, 4)), Less(Abs(tau), Div(1, 4)), Or(Unequal(k, 0), Greater(Abs(z), 1))), Equal(Subscript(L, 1), Add(Log(z), Mul(Mul(Mul(2, ConstPi), ConstI), k))), Equal(Subscript(L, 2), Log(Subscript(L, 1))), Equal(sigma, Div(1, Subscript(L, 1))), Equal(tau, Div(Subscript(L, 2), Subscript(L, 1))))))

Topics using this entry

Lambert W-function