Does the following have a solution?

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Thread starter Frogeyedpeas
Start date Mar 20, 2012

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Frogeyedpeas

Thread starter
Mar 20, 2012

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

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JJacquelin

Mar 21, 2012

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.

Does the following have a solution?

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Does the following have a solution?

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