Branch point

A point near which analytic continuation around a loop can permute the local branches of a multivalued analytic function.

Analytic continuation: The extension of an analytic function along a path through overlapping domains. Continuation along loops encircling a branch point can return a different function value.
Holomorphic function: A complex-valued function that is complex differentiable throughout an open domain. Branch behavior concerns attempts to extend holomorphic functions across or around excluded points.
Complex square root: A multivalued complex function whose values square to the input. A loop around zero exchanges its two local values.
Branch cut: A curve or set removed from a domain to choose a single-valued branch of a multivalued function. Cuts are conventional barriers drawn from branch points, not intrinsic changes to the function.
Pole (complex analysis): An isolated singularity at which a meromorphic function diverges like a finite negative power. A pole has a single-valued Laurent expansion, unlike a branch point’s permuted values.
Monodromy: The transformation of analytic-function branches induced by continuation around closed paths. Its permutation of branches is the defining behavior measured around this point.
Multivalued function: A rule assigning several possible values to an input, often arising from inverse analytic functions. Its local values form the branches that continuation can permute.
Complex logarithm: A multivalued inverse of the complex exponential, differing by integer multiples of 2πi. A loop around zero adds 2πi, producing infinitely many branches.
Singularity of a complex function: A point where a complex function is not holomorphic, though it may have different types of local behavior. Branch points are singularities that cannot be made removable or poles by choosing a local branch.
Essential singularity: An isolated singularity that is neither removable nor a pole. Its wild local behavior differs from the structured permutation of branches at a branch point.

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