1550251515-Classical_Complex_Analysis__Gonzalez_

"

Integration

y

B (2+i)

0 A^3 x

Fig. 7.2

7 .4 INTEGRAL OF A COMPLEX FUNCTION OF A

COMPLEX VARIABLE ALONG A PIECEWISE

DIFFERENTIABLE ARC

415

Definition 7.3 can be extended to a piecewise differentiable arc as follows.

Definition 7.4 Let 1: z = z(t), a::; t::; (3, be a piecewise differentiable

arc, and let f(z) be a continuous function along 'Y· If r1, r 2 , ... , 'Tk are·

the points of (a, (3] where the derivative z'(t) has discontinuities of the first

kind, we set

J f(z)dz= 1ri J(z(t))z'(t)dt+ ... + 1(3 J(z(t))z'(t)dt

7 k

(7.4-1)

where z(t) may be given by different equations in different subarcs. This

definition allows us to take and integral along a polygonal line as well as

along other continuous arcs with "corner" points.

Example Consider the integral J z dz, where 'Y is the polygonal line OAB
of Fig. 7.2. We have

7

,

OA: z = t,

AB: z = 2+ it,

Hence

j z dz = fo

2

t dt + fo\2 -it)i dt = % + 2i

7

7.5 OTHER TYPES OF COMPLEX INTEGRALS

Definitions 7.5 Let f(z) be a continuous complex function along the

continuously differentiable arc 1: z = z(t) = x(t) + iy(t), a ::; t :::; (3.