A branch point of an analytic function is a point in the complex plane whose complex argument can be mapped from a single point in the domain to multiple points in the range. For example, consider the behavior of the point z = 0 under the power function f(z) = z^a for complex non-integer a, i.e., a element C with a not element Z. Writing z = e^(i θ) and taking θ from 0 to 2π gives f(e^(0i)) | = | e^0 = 1 f(e^(2π i)) | = | e^(2π i a), so the values of f(z) at arg(z) = 0 and arg(z) = 2π are different, despite the fact that they correspond to the same point in the domain.

branch | branch cut | logarithmic singularity | pinch point