We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy’s theorem and Cauchy’s integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy’s integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions.
- 8.1: Geometric Series
- Having a detailed understanding of geometric series will enable us to use Cauchy’s integral formula to understand power series representations of analytic functions. We start with the definition:
- 8.2: Convergence of Power Series
- When we include powers of the variable z in the series we will call it a power series. In this section we’ll state the main theorem we need about the convergence of power series. Technical details will be pushed to the appendix for the interested reader.
- 8.3: Taylor Series
- The previous section showed that a power series converges to an analytic function inside its disk of convergence. Taylor’s theorem completes the story by giving the converse: around each point of analyticity an analytic function equals a convergent power series.
- 8.4: Taylor Series Examples
- The uniqueness of Taylor series along with the fact that they converge on any disk around z0 where the function is analytic allows us to use lots of computational tricks to find the series and be sure that it converges.
- 8.5: Singularities
- A function f(z) is singular at a point z0 if it is not analytic at z0
- 8.7: Laurent Series
- The Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
- 8.9: Poles
- Poles refer to isolated singularities.
Thumbnail: A Laurent series is defined with respect to a particular point \(c\) and a path of integration \(γ\). The path of integration must lie in an annulus, indicated here by the red color, inside which f(z) is holomorphic (analytic). (Public Domain; Pko via Wikipedia)