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Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.

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This equation cannot be solved in terms of the Lambert $W$ function or any other known functions -- even if a, r1, and r2 are real, let alone complex.

Here is what Mathematica says about this:

enter image description here

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    $\begingroup$ Lambert W works when r1, and r2 are real. $\endgroup$
    – Hamed Elwarfalli
    Commented Feb 5 at 18:39
  • $\begingroup$ @HamedElwarfalli : How does it work then, for real r1 and r2? I believe you are mistaken. $\endgroup$
    – Iosif Pinelis
    Commented Feb 5 at 18:43
  • $\begingroup$ mathoverflow.net/questions/217499/… $\endgroup$
    – Hamed Elwarfalli
    Commented Feb 5 at 19:18
  • $\begingroup$ @HamedElwarfalli : Looking at those linked papers, it appears that the function "solving" the equation was just given a name (of a generalized Lambert function), and some properties of such a function have been studied, including a formal series representation. However, even the radius of convergence of that representation has not been determined; see also p. 17 (perhaps, the radius of convergence is $0$). Apparently, this is why Mathematica does not consider these developments notable. $\endgroup$
    – Iosif Pinelis
    Commented Feb 5 at 19:59
  • $\begingroup$ @HamedElwarfalli : And, of course, the genuine Lambert $W$ function does not solve your equation. for real or complex r1 and r2 (unless r1=r2). $\endgroup$
    – Iosif Pinelis
    Commented Feb 5 at 21:35

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