Kheyfits, Alexander I. Closed-form representations of the Lambert \(W\) function. (English) Zbl 1077.33036 Fract. Calc. Appl. Anal. 7, No. 2, 177-190 (2004). 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 \(z\). Cited in 2 Documents 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 Keywords:Lambert \(W\) function; special functions; explicit solutions of transcendental equations PDFBibTeX XMLCite \textit{A. I. Kheyfits}, Fract. Calc. Appl. Anal. 7, No. 2, 177--190 (2004; Zbl 1077.33036) Digital Library of Mathematical Functions: §4.13 Lambert 𝑊-Function ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions