Complex Logarithm and Branch Cuts

Core Idea

Since e^z is periodic, the logarithm is multi-valued: log(w) = log|w| + i(arg(w) + 2πk) for any integer k. To make log single-valued, we choose a branch cut (conventionally the negative real axis) and define a principal branch Log(w). The principal logarithm Log is holomorphic on ℂ minus the cut and satisfies (Log(z))' = 1/z.

How It's Best Learned

Trace a path around the origin in the complex plane and observe how Log(z) changes; this reveals the branch cut and the multi-valuedness of the logarithm. Compare the principal branch with other branches.

Common Misconceptions

Thinking log is single-valued like the real logarithm; all branches are equally valid. Assuming the branch cut is arbitrary; while the location is arbitrary, the fact that a cut is needed is not.

Explainer

You already know the complex exponential e^z, and one of its defining features is that it is periodic: e^{z + 2πi} = e^z for every z. This means infinitely many inputs map to the same output. For the real exponential, e^x is strictly increasing and therefore injective — each output comes from exactly one input, so the real logarithm is well-defined. The complex exponential's periodicity destroys injectivity and forces the complex logarithm to be multi-valued: if log(w) = z, then z + 2πi, z + 4πi, z − 2πi, and so on are all equally valid logarithms of w.

To see this concretely, write w in polar form as w = r·e^iθ. Then e^z = w requires e^{Re z}·e^{i Im z} = r·e^iθ, giving Re(z) = log r (the real logarithm of the modulus) and Im(z) = θ + 2πk for any integer k. So log(w) = log|w| + i·arg(w), where arg(w) can take any of infinitely many values differing by multiples of 2π. There is no canonical choice — they are all legitimate.

To use the logarithm in analysis — to integrate 1/z, to define complex powers z^α — we need a single-valued function. The solution is to choose a branch cut: a curve from the origin to infinity that we agree never to cross. The standard choice is the negative real axis. By declaring that we always measure the argument in (−π, π], we select exactly one angle for each nonzero complex number (except those on the cut itself, where the function is left undefined). This gives the principal branch, written Log(z), which is holomorphic on ℂ minus the negative real axis and satisfies (Log z)′ = 1/z.

The key conceptual point is that the cut is a mathematical choice, not a physical one — a different cut (say, the positive imaginary axis) would give a different branch, equally valid, holomorphic on a different domain. What is not a choice is that some cut is necessary: trying to make the logarithm continuous everywhere on ℂ \ {0} is impossible, because any loop around the origin forces the argument to increase by 2π, creating a discontinuity somewhere. The branch cut marks exactly where that unavoidable discontinuity lives.