ch10 - Chapter Ten Poles Residues and All That 10.1...

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Chapter Ten Poles, Residues, and All That 10.1. Residues. A point z 0 is a singular point of a function f if f not analytic at z 0 , but is analytic at some point of each neighborhood of z 0 . A singular point z 0 of f is said to be isolated if there is a neighborhood of z 0 which contains no singular points of f save z 0 . In other words, f is analytic on some region 0 | z z 0 |   . Examples The function f given by f z 1 z z 2 4 has isolated singular points at z 0, z 2 i , and z 2 i . Every point on the negative real axis and the origin is a singular point of Log z , but there are no isolated singular points. Suppose now that z 0 is an isolated singular point of f . Then there is a Laurent series f z j  c j z z 0 j valid for 0 | z z 0 | R , for some positive R . The coefficient c 1 of z z 0 1 is called the residue of f at z 0 , and is frequently written z z 0 Res f . Now, why do we care enough about c 1 to give it a special name? Well, observe that if C is any positively oriented simple closed curve in 0 | z z 0 | R and which contains z 0 inside, then c 1 1 2 i C f z dz . 10.1
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This provides the key to evaluating many complex integrals. Example We shall evaluate the integral C e 1/ z dz where C is the circle | z | 1 with the usual positive orientation. Observe that the integrand has an isolated singularity at z 0. We know then that the value of the integral is simply 2 i times the residue of e 1/ z at 0. Let’s find the Laurent series about 0. We already know that e z j 0 1 j ! z j for all z . Thus, e 1/ z j 0 1 j ! z j 1 1 z 1 2! 1 z 2  The residue c 1 1, and so the value of the integral is simply 2 i . Now suppose we have a function f which is analytic everywhere except for isolated singularities, and let C be a simple closed curve (positively oriented) on which f is analytic. Then there will be only a finite number of singularities of f inside C (why?). Call them z 1 , z 2 , , z n . For each k 1,2, , n , let C k be a positively oriented circle centered at z k and with radius small enough to insure that it is inside C and has no other singular points inside it.
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