Does the following have a solution?

[Frogeyedpeas]

Does the equation x^x = -i have a solution in the complex plane?

 

[JJacquelin]

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.