Page 38 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 4: Integration

Figure 4.1 Notes

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 , z , z ,....., z } is a finite subset of C, such that a = z , b = z , and such that z j
2
0
1
n
0
n
comes immediately after z as we travel along C from a to b.
j–1
A Riemann sum associated with the partition P is just what it is in the real case:
n
*
S(P)   f(z ) z ,

j
j
j 1

where z is a point on the arc between z and z, and zj = z – z .
*
j–1
j
j
j–1
j
Notes For a given partition P, there are many S(P)—depending on how the points z * j
are chosen.)
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.


 (f(z) g(z))dz   f(z)dz   g(z)dz
C C C
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 < t < t 2
1
0

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