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Zbl 1077.33036
Kheyfits, Alexander I.
Closed-form representations of the Lambert $W$ function.
(English)
[J] Fract. Calc. Appl. Anal. 7, No. 2, 177-190 (2004). ISSN 1311-0454; ISSN 1314-2224/e

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 2000:
*33F05 Numerical approximation of special functions
33B99 Elementary classical functions
65H05 Single nonlinear equations (numerical methods)
30D99 Entire and meromorphic functions

Keywords: Lambert $W$ function; special functions; explicit solutions of transcendental equations

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