Exercises on the Lambert W Function

1. Solve the following transcendental equations for the indicated variables, in terms of W.
   1. for y
   2. for x
   3. for w
   4. , , for .
   5. , , for
   6. for
   7. for x
   8. for x
   9. for p
   10. for y.
2. Show that Euler's series for ,

   

   converges for all v small enough that

   

   where is the Euclidean mean of a and b.

3. Iterated exponentiation.
   1. If

      

      what is x? (High School problem)

   2. If

      

      what is x? Explain, comparing to the previous problem. (2nd year analysis question)

   3. Find a closed-form expression for the tower of exponentials in the previous questions, in terms of W.
4. Show that

   

   where the polynomials satisfy and

   

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