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

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

View Full Document Right Arrow Icon
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 ) = X n =0 a n ( z - z 0 ) n + X n =1 b n ( z - z 0 ) n . In particular b 1 = 1 2 πi Z C f ( z ) dz, 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
where C is any positively oriented, simple closed contour which lies in D and
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 11/23/2009 for the course MATHEMATIC Mathematic taught by Professor Dansloughter during the Spring '04 term at Furman.

Page1 / 4

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

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