LambertW

The Lambert W function, also called the omega function or product logarithm.

lambertw(z,k)   # Lambert W function for argument z and branch index k
lambertw(z)     # the same as lambertw(z,0)

z may be Complex or Real. k must be an integer. For Real z, k must be either 0 or -1.

Examples:

julia> lambertw(10)
1.7455280027406994

julia> lambertw(e)
1

julia> lambertw(1.0)
0.5671432904097838

julia> lambertw(-pi/2 + 0im)  / pi
4.6681174759251105e-18 + 0.5im

Return 1 + W(-1/e + z), for z satisfying 0 <= abs(z) < 1/e, on the branch of index k, where k must be either 0 or -1. This function is designed to minimize loss of precision near the branch point z=-1/e. lambertwbp(z,k) converges to Float64 precision for abs(z) < 0.32.

If k=-1 and imag(z) < 0, the value on the branch k=1 is returned.

julia> lambertwbp(1e-3,-1)
-0.07560894118662498

julia> lambertwbp(0)
-0.0

lambertwbp uses a series expansion about the branch point z=-1/e. The loss of precision in lambertw is analogous to the loss of precision in computing the sqrt(1-x) for x close to 1.

The functional inverse of the Lambert W function.

julia> finv(lambertw)(lambertw(1.0))
1.0

julia> finv(lambertw)(lambertw(1+im/2,3))
1.0 + 0.49999999999999944im

The omega constant

julia> ω
ω = 0.5671432904097...

julia> omega
ω = 0.5671432904097...

julia> ω * exp(ω)
1.0

julia> big(ω)
5.67143290409783872999968662210355549753815787186512508135131079223045793086683e-01 with 256 bits of precision

julia> lambertw(1) == float(ω)
true