Complex logarithm

In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the real natural logarithm ln x is the inverse of the real exponential function e^x. Thus, a logarithm of a complex number z is a complex number w such that e^w = z. The notation for such a w is ln z or log z. Since every nonzero complex number z has infinitely many logarithms, care is required to give such notation an unambiguous meaning.

If z = re^iθ with r > 0 (polar form), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2πi gives all the others.

For a function to have an inverse, it must map distinct values to distinct values, i.e., be injective. But the complex exponential function is not injective, because e^w+2πi = e^w for any w, since adding iθ to w has the effect of rotating e^w counterclockwise θ radians. So, the sequence of equally spaced points along a vertical line

are all mapped to the same number by the exponential function, which means that the exponential function does not have an inverse function in the standard sense. There are two solutions to this problem.

One is to restrict the domain of the exponential function to a region that does not contain any two numbers differing by an integer multiple of 2πi: this leads naturally to the definition of branches of log z, which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of arcsin x on [−1, 1] as the inverse of the restriction of sin θ to the interval [−π/2, π/2]: there are infinitely many real numbers θ with sin θ = x, but one arbitrarily chooses the one in [−π/2, π/2].

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