Laurent series

By definition, the Laurent series is a power series centred at a point $z=z_0$ in the form:

$$f(z)=\sum _{k=-\infty }^{\infty }{a}_k(z-z_0{)}^k$$ $$f(z)=\cdots +\frac{a_{-3}}{(z-z_0{)}^3}+\frac{a_{-2}}{(z-z_0{)}^2}+\frac{a_{-1}}{z-z_0}+a_1(z-z_0)+\cdots$$

The idea with the Laurent series is that a complex-valued function may fail to be analytic at a given point (i.e., it has a singularity), but we can still find a power series to expand around that point. We study Laurent series with the goal of being able to compute complicated integrals.

Terminology and theorems

For a Laurent series:

$$\sum _{n=-\infty }^{\infty }{a}_n(z-z_0{)}^n=\sum _{-\infty }^{-1}{a}_n(z-z_0{)}^n+\sum _{n=0}^{\infty }{a}_n(z-z_0{)}^n$$

We call the first term, with the negative powers of $n$ the principal part. The second term, with positive powers of $n$, is called the analytic part.

Laurent's theorem

Suppose $f(z)$ is analytic in a region containing the annulus $r<|z-z_0|<R$ then it can be expanded in a Laurent series ${\sum }_{n=-\infty }^{\infty }{a}_n(z-z_0{)}^n$ which is absolutely convergent for $r<|z-z_0|<R$.

The $a_{-1}$ term is called the residue of $f(z)$ at $z_0$. This ends up being a pretty important concept — keep an eye on it.

Computations

Finding the Laurent series is much less straightforward than finding the Taylor series, especially if its centred at a different point than we expect.

Here’s a set of steps:

• Where we’re able, find the Taylor expansion centred at a given point.
• Where we’re not, we should expand certain terms and multiply back.
  • i.e., $z\to z+2-2$
  • This doesn’t make intuitive sense, but we can multiply out one of the terms so that we get a series truly centred at a certain point.