Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2013 / Analysis / 6 / a

An isolated singularity at $a$ means that $f$ is holomorphic in some punctured disk $0<|z-a|<r$, while $a$ itself is an excluded point or a point where the given function is not holomorphic. There are no other singularities arbitrarily close to $a$. By Laurent's theorem, the function has a unique Laurent series on that punctured disk:The negative-power terms form its principal part of a Laurent series.

A removable singularity is an isolated singularity for which the function on the punctured disk extends to a holomorphic function on the full disk, by assigning or correcting its value at $a$. In Laurent series terms,The remaining power series defines the extension with value $c_0$ at $a$. Conversely, a holomorphic extension has a Taylor series there, so uniqueness of the Laurent series forces all negative coefficients to vanish.

A pole of order $m\ge 1$ at $a$ has finitely many negative powers in its Laurent series, the lowest being exactly $-m$:Equivalently,Thus $(z-a{)}^{m}f(z)$ has a holomorphic extension with nonzero value at $a$, whereas multiplying by any smaller power does not remove the singularity.

A zero of order $n\ge 1$ at $a$ means that $f$ is holomorphic near $a$ andEquivalently its Taylor series starts with the nonzero coefficient of $(z-a{)}^n$, soUnlike a pole, a zero is a regular point of the function itself; its reciprocal has a pole of the corresponding order.