Cauchy integrals and the Laurent series

Theorem B.5 If f z) is analytic in some simply connected domain D, the integral over f (2) along some closed path C pertaining to D vanishes. 

Using Gaufi s theorem [see Eg. (B.27)] we may rewrite the original integral 

As f (2) IS analytic, the Cauchy-Riemann differential equations are satisfied so that each of the two integrals above vanishes identically regardless of the specific choice of the path C. q.e.d. 

Consider now a domain D, which is no longer simply connected. Such a situation arises if / (2) is anedytic everjnvhere in D except at some point 2 = 2] where / (2) is supposed to have a singularity (see Fig. B.2). Then, in fact, the integral around any closed path in D. surrounding the. singularity does not vanish but one may instead define the so-called residue 

Suppose now / (2) is analytic across a simply connected domain then it is immediately clear that, if we pick a point 2 = 20 in that domain, the quantity / (2) / (2 — 20) will have a singularity at that point. Because of the above, the integral over / (2) / (2 — 20) 2ilong any closed path surromiding 2 = 2o will have some nonzero value that we seek to calculate. Because the closed path C surrounding 2 = 20 is arbitrary, we take it to be a circle of 

The Laurent expansion is very useful in analyzing the nature of singularities. However, a discussion of this aspect would go way beyond the scope of this Appendix. Therefore, we emphasize only the relation of the Laurent series 

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