1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

6.2 • CONTOURS AND CONTOUR I NTEGRALS 201

y

• I + i I + i

Figure 6 .4 The polygonal path C = C1 + Cz + C3 from -1 + i to 3 -i.

Similarly, the segments C2 and C3 are given by

C2: z2(t)=(-l+2t)+it,

C3: Z3(t) = (1+2t) +i(l -2t)'

for 0 ::; t ::; 1, and

for 0::; t::; 1.

We are now ready to define the integral of a complex function along a
contour C in the plane with initial point A and terminal point B. Our ap-

proach is to mimic what is done in calculus. We create a partition Pn =

{ zo = A, z1, z2, ... , z.. = B} of points that proceed along C from A to B and

form the differences 6.zk = Zk - Zk-i, fork= 1, 2, ... , n. Between each pair of

partition points Zk- I and zk we select a point ck on C, as shown in Figure 6.5,
and evaluate the function f. We use these values to make a Riemann sum for
the partition:
n n
S (Pn) = L f (ck) (zk - Zk-1) = L f (ck) 6.zk· (6-12)
k=I
Assume now that there exists a unique complex number L that is the limit
of every sequence {S (Pn)} of Riemann s ums given in Equation (6-12), where the
maximum of Jb.zkJ tends toward 0 for the sequence of partitions. We define the
number L as the value of the integral of the function f taken along the contour
c.
y

zo=A

-+-------------~x

Figure 6.5 Partition points {zk} and function evaluation points {ck} for a Riemann

sum along the contour C from z =A t.o z = B.