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7 Taylor and Laurent series - MIT Mathematics

Topic 7 NotesJeremy Orloff7 Taylor and Laurent IntroductionWe originally defined an analytic function as one where the derivative, defined as a limitof ratios, existed. We went on to prove Cauchy s theorem and Cauchy s integral revealed some deep properties of analytic functions, the existence of derivativesof all goal in this topic is to express analytic functions as infinite power series . This will leadus to Taylor series . When a complex function has an isolated singularity at a point we willreplace Taylor series by Laurent series . Not surprisingly we will derive these series fromCauchy s integral we come to power series representations after exploring other properties of analyticfunctions, they will be one of our main tools in understanding and computing with geometric seriesHaving a detailed understanding of geometric series will enable us to use Cauchy s integralformula to understand power series representations of analytic functions.

The geometric series is so fundamental that we should check the root test on it. Example 7.4. Consider the geometric series 1 + z+ z2 + z3 + :::. The limit of the nth roots of the terms is L= lim n!1 jznj1=n= limjzj= jzj Happily, the root test agrees that the geometric series converges when jzj<1. 7.4 Taylor series

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