1550251515-Classical_Complex_Analysis__Gonzalez_

412 Chapter^7

B -----------

b

g

a -----

A

0 c d T

Fig. 7.1

lb v(t)dt = 1d v[g(r)]g'(r)dr

By adding to the first equation the second multiplied by i, we get

lb f(t)dt = 1df[g(r)]g'(r)dr

To prove 9, let F(t) = U(t) + iv(t). Then F'(t) = f(t) for a ::; t ::; b
implies that U'(t) = u(t) and V'(t) = v(t) for a ::; t ::; b. Hence, by the
fundamental theorem of real integral calculus, we have

lb f(t)dt =lb u(t)dt+i lb v(t)dt

= U(b)-U(a) + i[V(b) - V(a)]

= [U(b) + iV(b)] - [U(a) + iV(a)]

= F(b)-F(a)

This result shows that the so-called fundamental theorem of integral cal-

culus, which allows the evaluation of the definite integral of f by means

of a primitive function F, is also valid for complex continuous functions

of a real variable..

Property 10 means that the definite integral of f with a variable upper

lin{it defines a primitive function off .. Again, the property follows easily
from the corresponding property for real integrals. In fact, if we let

' t t

G(t)=g(t)+ih(t)= la u(r)dr+i}a v(r)dr