19.1 Sequences and Series

19.2 Taylor Series

19.3 Laurent Series

19.4 Zeros and Poles

19.5 Residue Theorem

19.6 Evaluation of Real Integrals

Chapter 19 in Review

Cauchy’s integral formula for derivatives indicates that if a function f is analytic at a point z₀, then it possesses derivatives of all orders at that point. As a consequence, we will see that f can always be expanded in a power series at that point. On the other hand, if f fails to be analytic at z₀, then we may still be able to expand it in a different kind of series known as a Laurent series. The notion of Laurent series leads to the concept of a residue, and this, in turn, leads to yet another way of evaluating complex integrals.