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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

Page1 / 9

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online