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Does the following have a solution?

  1. Mar 20, 2012 #1
    Does the equation x^x = -i have a solution in the complex plane?
  2. jcsd
  3. Mar 21, 2012 #2
    z^z = -i
    exp(ln(z)*z) = -i = exp(i*pi(4n-1)/2)
    z*ln(z) = i*pi*(4n-1)/2
    z =exp(Z)
    Z*exp(Z) = i*pi*(4n-1)/2
    Z = W(i*pi*(4n-1)/2)
    W is the Lambert's W function.
    z = exp(w(i*pi*(4n-1)/2))
    W is a multivalued function. One have to consider all branches.
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