Zeros and Singularities

Zeros and Singularities

A point $z_0$ is called a zero of an order $m$ of a complex function $f$ if $f$ is analytic at $z_0$ and

\[0 = f(z_0) = f'(z_0) = \cdots = f^{(m-1)}(z_0), f^{(m)}(z_0) \neq 0.\]

Zeroes of order 1 are often called simple zeros.

Theorem: if a function $f(z)$ can be expressed in the form

\[f(z) = (z - z_0)^m g(z)\]

valid in some circle $|z - z_0| < R$, where $g(z)$ is analytic at $z_0$ and $g(z_0) \neq 0$, then $f(z)$ has a zero of order $m$ at $z_0$.

A point $z_0$ is called a singularity of a complex function $f$ if $f$ is not analytic at $z_0$, but every neighborhood of $z_0$ contains at least one point at which $f$ is analytic.

A singularity $z_0$ of a complex function $f$ is said to be isolated if there exists a neighborhood of $z_0$ in which $z_0$ is the only singularity of $f$.