Fungrim entry: da0f15

\left|W_{k}\!\left(z\right) - \left(\operatorname{L_1} - \operatorname{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 } \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:

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| < \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:

\left|W_{k}\!\left(z\right) - \left(\operatorname{L_1} - \operatorname{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 } \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}}

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| < \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
`Abs`	$\left\|z\right\|$	Absolute value
`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
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	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(z, k), Parentheses(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, Sub(M, 1))), For(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(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, 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(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

Lambert W-function