# Expanding in powers of 1/z (Laurent series)

## Main Question or Discussion Point

The text book used in one of my courses talks about expanding functions in powers of 1/z aka negative powers of z.

The problem is that I cannot recall that any previous course taught me/challenged me on how to expand functions in negative powers. For example, Taylor series only have positive powers.

Is there a general method of expanding in negative powers, like for Taylor series, or are there at best similar methods for similar functions?

I fear I have overlooked something elementary here, because I feel strangely clueless about this one (and Internet searches have made me no wiser). The textbook only gives examples of the results of expansion in 1/z, but never gives any details on how it is done.

Office_Shredder
Staff Emeritus
Gold Member
Is this a complex analysis course, or is it something that had complex analysis as a prerequisite?

Practically, to calculate these you can often do standard Taylor series calculations
$$f(x)=\frac{x}{1-x} = \frac{1}{1-1/x}$$

we know how to expand 1/(1-1/x) using the Taylor series for 1/(1-x)
$$f(x) = 1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+...$$
and this is valid as long as |x|>1

It's a physics course without physics, if that makes any sense. To answer the question though, complex analysis is a part of the course rather then a prerequisite (e.g. it includes the most basic proofs/definitions for differention of functions of a complex variable).

I do recall seeing the Taylor expansion you introduced (in an introductory course in astropysics, as a matter of fact).

Anyway, I guess my question has been answered.