Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined or lacks some "nice" property, such as differentiability.

For example, the function f(x) = 1/x has a singularity at x = 0, where it explodes to ±∞ and isn't defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it isn't differentiable there. The algebraic set defined by y² = x² in the (x,y) coordinate system has a singularity at (0,0) because it doesn't admit a tangent there. The algebraic set defined by y² = x also has a singularity at (0,0), this time because it has a "corner" at that point.

In complex analysis, we distinguish three types of singularities. Suppose U is an open subset of C, a is an element of U and f is a holomorphic function 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 point a is a pole 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 point a is an essential singularity of f if it is neither a removable singularity nor a pole.