Solve the following equation with Wright omega function?

Questions
1

Need your help please

I’m trying to solve the following equation for the variable R:

(aW(R))^(1/(1-a))+(aW(R))^(a/(1-a))=P by using the following code

Solve[{(a ProductLog[R])^(1/(1 - a)) + (a ProductLog[ R])^(a/(1 - a)) == P}, R]

here “W” is the wright omega function (or Lamber function), “a” is a constant varying from 0.5 to 0.65 and P is a constant.

When putting a =0.5, a closed from solution is found easily, but when trying to generate solution for each value of “a”, Wolfram Mathematica still running without converging.

Any idea plz?

I need your help.

Thanks in advance.

2

I doubt there’s a closed form for the equation for general constant a and P.

a = 65/100; P = 1;
Solve[{(a ProductLog[R])^(1/(1 - a)) + (a ProductLog[R])^(a/(1 - a)) == P}, R, Reals]

(* {{R -> E^Root[{-13 + #1^7 &, -5 + #2^7 &, -2 + #3^7 &, 
      800 #1 #2^6 #3^5 - 3380 #4^13 - 2197 #4^20 &}, {1, 1, 1, 
      2}]^7 Root[{-13 + #1^7 &, -5 + #2^7 &, -2 + #3^7 &, 
      800 #1 #2^6 #3^5 - 3380 #4^13 - 2197 #4^20 &}, {1, 1, 1, 2}]^7}} *)

%//N
{{R -> 3.57197}}
3

thnks for your response

but i m looking for a generalized solution function of “a” and the constant “P”

i have the same doubt that there is no a closed form for general constant a and P

But who knows, that is why i m asking your help

Best regards

4

I doubt there is any general form. A lesser but related problem would be to handle the equation r+r^a+b==0 and solve for r. Even for concrete values of b I do not know what an analytic solution might be.