In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is ln z. But because every nonzero complex number z has infinitely many logarithms, care is required to give this notation an unambiguous meaning.
If z = reiθ with r > 0 (polar form), then w = ln (reiθ ), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2πi gives all the others.
Read more about Complex Logarithm: Problems With Inverting The Complex Exponential Function, Definition of Principal Value, Branches of The Complex Logarithm, The Complex Logarithm As A Conformal Map, Applications, Plots of The Complex Logarithm Function (principal Branch)
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