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Closed-form representations of the Lambert \(W\) function. (English) Zbl 1077.33036

Summary: The Lambert \(W\) function is the many-valued analytic inverse of \(z(w)= we^w\). We use elementary complex analysis to derive closed-form representations of all of the branches of \(W\) through simple quadratures. For instance, if \(-\pi<\arg z<\pi\), then the \(k\)th, \(k= 0,\pm1,\pm2,\dots\), branch of \(W\) is given by \[ W_k(z)= {\int^\infty_0 {xdx\over (x^2+ 1)B}+ {\ln z+ 2k\pi i\over (\ln z+ 2k\pi i)^2+ (\pi/2+ 1)^2}\over {\pi/2+1\over (\ln z+ 2k\pi i)^2+ (\pi/2+ 1)^2} -\int^infty_0 {dx\over (x^2+ 1)B}}, \] where \(B= (x- \ln x+ \ln z+ 2k\pi i)^2+ \pi^2\). A similar expression holds true for negative \(z\).

MSC:

33F05 Numerical approximation and evaluation of special functions
33B99 Elementary classical functions
65H05 Numerical computation of solutions to single equations
30D99 Entire and meromorphic functions of one complex variable, and related topics
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