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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 -π<argz<π, then the kth, k=0,±1,±2,, branch of W is given by

W k (z)= 0 xdx (x 2 +1)B+lnz+2kπi (lnz+2kπi) 2 +(π/2+1) 2 π/2+1 (lnz+2kπi) 2 +(π/2+1) 2 - i nfty 0 dx (x 2 +1)B,

where B=(x-lnx+lnz+2kπi) 2 +π 2 . A similar expression holds true for negative z.

33F05Numerical approximation and evaluation of special functions
33B99Elementary classical functions
65H05Single nonlinear equations (numerical methods)
30D99Entire and meromorphic functions