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# lecture-44 - Lecture 44 Residues Dan Sloughter Furman...

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Lecture 44: Residues Dan Sloughter Furman University Mathematics 39 May 21, 2004 44.1 Some terminology Recall that we say a point z 0 is a singular point of a function f if f is not analytic at z 0 but is analytic at some point in every neighborhood of z 0 . We will say that z 0 is an isolated singular point if it is a singular point and there exists > 0 such that f is analytic in the deleted neighborhood 0 < | z - z 0 | < . Example 44.1. Both z = i and z = - i are isolated singular points of f ( z ) = 1 1 + z 2 . Example 44.2. z = 0 is a singular point, but not an isolated singular point, of f ( z ) = Log( z ). If z 0 is an isolated singular point of f , then there exists R > 0 such that f is analytic in D = { z C : 0 < | z - z 0 | < R } . It follows that f ( z ) has a Laurent series representation for all z D : f ( z ) = n =0 a n ( z - z 0 ) n + n =1 b n ( z - z 0 ) n . In particular b 1 = 1 2 πi C f ( z ) dz, 1

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where C is any positively oriented, simple closed contour which lies in D and has z 0 in its interior. In other words, C f ( z ) dz = 2 πib 1 .
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lecture-44 - Lecture 44 Residues Dan Sloughter Furman...

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