Mathematical Singularity - Complex Analysis

Complex Analysis

In complex analysis there are four classes of singularities, described below. Suppose U is an open subset of the complex numbers C, and the point a is an element of U, and f is a complex differentiable function defined on some neighborhood around a, excluding a: U \ {a}.

• Isolated singularities: Suppose the function f is not defined at a, although it does have values defined on U \ {a}.
  • The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U \ {a}. The function g is a continuous replacement for the function f.
  • The point a is a pole or non-essential singularity of f if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (z − a)n for all z in U \ {a}. The derivative at a non-essential singularity may or may not exist. If g(a) is nonzero, then we say that a is a pole of order n.
  • The point a is an essential singularity of f if it is neither a removable singularity nor a pole. The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.
• Branch points are generally the result of a multi-valued function, such as or being defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The location and shape of most of the branch cut is usually a matter of choice, with perhaps only one point (like for ) which is fixed in place.