Advanced Engineering Mathematics, 6th Edition

CHAPTER 19: Series and Residues

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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 of this result, 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 a point 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.

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