Message Boards Message Boards

GROUPS:

Solve the following equation with Wright omega function?

Posted 1 month ago
283 Views
|
3 Replies
|
1 Total Likes
|

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.

3 Replies

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}}

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

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.

Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract