Complex Logarithm - Problems With Inverting The Complex Exponential Function

Problems With Inverting The Complex Exponential Function

For a function to have an inverse, it must map distinct values to distinct values. But the complex exponential function does not have this injective property because: ew+2πi = ew for any w, since adding iθ to w has the effect of rotating ew counterclockwise θ radians. Even worse, the infinitely many numbers

forming a sequence of equally spaced points along a vertical line, are all mapped to the same number by the exponential function. So 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 sin−1x on as the inverse of the restriction of sin θ to the interval : there are infinitely many real numbers θ with sin θ = x, but one (somewhat arbitrarily) chooses the one in .

Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane, but a Riemann surface that covers the punctured complex plane in an infinite-to-1 way.

Branches have the advantage that they can be evaluated at complex numbers. On the other hand, the function on the Riemann surface is elegant in that it packages together all branches of log z and does not require any choice for its definition.