HP49-50G Lambert function

[Albert Chan]

I tried solving simultaneous equation for w = a+b*1j, using Tk! Solver.
Note: w * exp(w) = xr + 1j*xi

xr = exp(a) * (a*cos(b)-b*sin(b))
xi = exp(a) * (a*sin(b)+b*cos(b))

Rectangular form does not converge well, unless we gives it a good guess, for both a and b.
Converted to polar form, convergence is easier, only required guess of a = b = 0

x = w exp(w) = exp(w.real)*abs(w) * exp(1j*(w.imag + phase(w)))

TK Solver Rule Sheet

Code:

c = xr*xr + xi*xi  "abs(x)**2
d = atan2(xi,xr)   "phase(x)
a*a + b*b = c / exp(2*a)
mod(b+atan2(b,a)-d, 2*pi()) = 0