# ch4 - Chapter Four Integration 4.1 Introduction If D C is...

This preview shows pages 1–4. Sign up to view the full content.

Chapter Four Integration 4.1. Introduction. If : D C is simply a function on a real interval D , ,thenthe integral t dt is, of course, simply an ordered pair of everyday 3 rd grade calculus integrals: t dt x t dt i y t dt , where t x t iy t . Thus, for example, 0 1  t 2 1 it 3 dt 4 3 i 4 . Nothing really new here. The excitement begins when we consider the idea of an integral of an honest-to-goodness complex function f : D C ,where D is a subset of the complex plane. Let’s define the integral of such things; it is pretty much a straight-forward extension to two dimensions of what we did in one dimension back in Mrs. Turner’s class. Suppose f is a complex-valued function on a subset of the complex plane and suppose a and b are complex numbers in the domain of f . In one dimension, there is just one way to get from one number to the other; here we must also specify a path from a to b .Let C be a path from a to b , and we must also require that C be a subset of the domain of f . 4.1

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

View Full Document
Note we do not even require that a b ; but in case a b , we must specify an orientation for the closed path C . We call a path, or curve, closed in case the initial and terminal points are the same, and a simple closed path is one in which no other points coincide. Next, let P be a partition of the curve; that is, P z 0 , z 1 , z 2 , , z n is a finite subset of C , such that a z 0 , b z n , and such that z j comes immediately after z j 1 as we travel along C from a to b . A Riemann sum associated with the partition P is just what it is in the real case: S P j 1 n f z j z j , where z j is a point on the arc between z j 1 and z j ,and z j z j z j 1 . (Note that for a given partition P , there are many S P —depending on how the points z j are chosen.) If there is a number L so that given any  0, there is a partition P of C such that | S P L |  whenever P P , then f is said to be integrable on C and the number L is called the integral of f on C . This number L is usually written C f z dz . Some properties of integrals are more or less evident from looking at Riemann sums: C cf z dz c C f z dz for any complex constant c . 4.2
C f z g z  dz C f z dz C g z dz 4.2 Evaluating integrals. Now, how on Earth do we ever find such an integral? Let : , C be a complex description of the curve C . We partition C by partitioning the interval , in the usual way: t

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

### Page1 / 11

ch4 - Chapter Four Integration 4.1 Introduction If D C is...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online