☆ Π MTH ~CχχχVi~✧⁰¹³⁶ Π∞ßτ З³⁰¹ ⁰⁰כ
The calculation of Lambert's function \(W_0\) for
\[z\in\left[-\frac{1}{e},+\frac{1}{e}\right]\]
can be performed using the expression
\[W_0(z)=\sum_{n=1}^{\infty} \frac{(-n)^{n-1} }{n!}x^n\]
Some values obtained
\[\begin{matrix}
z&W_0(z)\\
\hline
-\frac{1}{e}&-0.931470\\
-0.30&-0.489402\\
-0.20&-0.259171\\
-0.10&-0.111832\\
0.00&+0.000000\\
0.10&+0.091276\\
0.20&+0.168915\\
0.30&+0.236755\\
\frac{1}{e}&+0.278590\\
\hline
\end{matrix}\]
It is important to note that at the extremes of the interval the values are compromised, let's look at the calculated values
\[\begin{matrix}
z&W_0(z)\\
\hline
-\frac{1}{e}&-1.000000\\
+\frac{1}{e}&+0.278464\\
\hline
\end{matrix}\]
On the other hand, for large values of z, something like \(z\gtrsim60\), we can use the formula
\[\begin{matrix}
W(z)=\\
\log{z}-\log{\log{z}}+\\
\sum_{k=0}^{\infty}\sum_{m=1}^{\infty} c_{km}(\log{\log{z}})^m(\log{z})^{-k-m}
\end{matrix}\]
Where
\[c_{km}=\frac{1}{m!}(-1)^k\begin{bmatrix}k+m\\k+1\end{bmatrix}\]
Remembering that the first-class Stirling number is
\[ \left[\begin{matrix}α\\β\end{matrix}\right]=\sum_{j=α}^{2α-β}(-1)^{j-β}\left(\begin{matrix}j-1\\β-1\end{matrix}\right)\left(\begin {matrix}2α-β\\j\end{matrix}\right)\left\{\begin{matrix}j-
β\\j-α\end{matrix}\right\}\]
And the second-class Stirling number is
\[\left\{\begin{matrix}α\\β\end{matrix}\right\}=\frac{1}{β!}\sum_{i=0}^α (-1)^i\left(\begin{matrix}β\\i\end{matrix}\right)(β-i)^α\]
A value obtained
\[\begin{matrix}
z&W_0(z)\\
\hline
70&3.112934\\
\hline
\end{matrix}\]
Using WolframAlpha, we obtain
\[ProductLog(0,70)\approx3.112930\]
αΩ
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