1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 447

Fig. 7.9

n-1

j f(z)dz = L j f(z)dz

'"'I k=O'"'lk

and if Fk is a primitive off on the disk Dk, we obtain

n-1

j f(z) dz= L[Fk(zk+1) - Fk(zk)]

'"'I k=O

(7.10-10)

This formula expresses the integral off along I in terms of local primitives

by decomposition of the arc I into a number of sufficiently small subarcs.

Remark Note that formula (7.10-10) requires the analyticity off but no
special property of I (except continuity). Hence (7.10-10) may be used to
define the integral of an analytic function along a continuous arc. This
observation is important for what follows.
To justify the definition of the integral by means of (7.10-10) we must
show that the right-hand side of the formula is independent of the choice
of the admissible sequences = {tk}· To do this, first we verify that the
value of the sum is unchanged if the sequence is refined by the insertion
of one or more additional terms. Clearly, it suffices to check this for the

insertion of just one term, say ti, such that tk :::; tj, < tk+l· If Di, is the

disk of center tk and radius E,zk = z(t'k) and Hk is the primitive off on

Dk, then on Dk n Dk we have (Fk - Hk)' = 0. Hence Fk -Hk is constant

in Dk n Dk, so (Fk - Hk)(zk+1) = (Fk - Hk)(zk), or Fk(zk+1) - Fk(zk) =

Hk(Zk+i) - Hk(zk), and it follows that

Hk(zZ) - Hk(zk) + Ih(zk+1) -Hk(zZ) = Fk(Zk+i) - Fk(zk)