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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,

t 2  1  it 3 dt 
0

1

4  i. 4 3

Nothingreally 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 thecomplex 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

Note we do not even require that a  b; but in case a  b, we must specify an orientation for theclosed 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 Riemannsum associated with the partition P is just what it is in the real case: SP 

 fz  z j , j
j1

n

where z  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 j given partition P, there are many SP—depending on how the points z  are chosen.) If j 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  fzdz.
C

Some properties of integrals are more or less evident from looking at Riemann sums:

 cfzdz  c  fzdz
C C

for any complex constant c.

4.2

fz  gzdz   fzdz   gzdz
C C C

4.2 Evaluating integrals. Now, how onEarth 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 

ft  t j   t j1 . j
j1

n

We have chosen the points z   t  , where t j1  t   t j . Next, multiply each term in the j j j sum by 1 in disguise:   ft   t jtj  tt j1  t j  t j1 . j j1
j1 n

SP 

I hope it is now reasonably convincing that ”in the limit”, we have


 fzdz   ft  tdt.
C 

(We are, of course, assumingthat the derivative   exists.)

Example We shall find the integral of fz  x 2  y  ixy from a  0 to b  1  i along three different paths, or contours, as some call them. First, let C 1 be the part of the parabola y  x 2 connecting the two points. A complex description of C 1 is  1 t  t  it 2 , 0  t  1:

4.3

1 0.8 0.6 0.4 0.2

0

0.2

0.4

x

0.6

0.8

1Now,  1 t  1  2ti, and f  1 t  t 2  t 2   itt 2  2t 2  it 3 . Hence,

 fzdz   f  1 t 1 tdt
C1 0

1

 2t 2  it 3 1  2tidt
0

1

 2t 2  2t 4  5t 3 idt
0

1

 4  5i 15 4 Next, let’s integrate along the straight line segment C 2 joining 0 and 1  i.
1 0.8 0.6 0.4 0.2

0

0.2

0.4

x

0.6

0.8

1

Here we have  2 t...