Does the following have a solution?

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SUMMARY

The equation x^x = -i has solutions in the complex plane, specifically through the use of the Lambert W function. By expressing z as exp(Z) and manipulating the equation to z*ln(z) = i*pi*(4n-1)/2, we can derive z = exp(W(i*pi*(4n-1)/2)). It is crucial to recognize that W is a multivalued function, necessitating consideration of all branches for comprehensive solutions.

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  • Knowledge of logarithmic properties in complex analysis
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Does the equation x^x = -i have a solution in the complex plane?
 
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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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