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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

Robert Corless
Tue Apr 7 11:26:19 EDT 1998