# Solve the following equation with Wright omega function?

Posted 6 months ago
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 Need your help pleaseI'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.
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Posted 6 months ago
 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}} 
 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.