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

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Chapter Four Integration 4.1. Introduction. If : D C is simply a function on a real interval D , , then the 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
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(Note we do not even require that a b ; but in case a b , we must specify an orientation for the closed path C .) 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
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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 0 t 1 t 2  t n . Then a , t 1 , t 2 , , b is partition of C . (Recall we assume that t 0 for a complex description of a curve C .) A corresponding Riemann sum looks like S P j 1 n f t j  t j t j 1
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