1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 413

we have

g(t) = lt u(r)dr, h(t) = lt v(r)dr

so that

g'(t) = u(t), h'(t) = v(t)

and

G'(t) = g'(t) + ih'(t) = u(t) + iv(t) = f(t)

To prove property 11, let f(t,s) = u(t,s) + iv(t,s) and F(s) = U(s) +

iV(s). Then

U(s) =lb u(t, s) dt, V(s) =lb v(t, s) dt

Since fs(t, s) is supposed to exist and be continuous on R, the partial

derivatives Us(t, s) and v 8 (t, s) also exist and are continuous on R. Hence,

by a known property of real integrals ([2], p. 219), we have

U'(s) =lb u 8 (t,s)dt, V'(s) =lb v 8 (t,s)dt

for s E (c, d]. Hence it follows that

F'(s) = U'(s) + iV'(s) =lb fs(t, s) dt

7 .3 INTEGRAL OF A COMPLEX FUNCTION OF A

COMPLEX VARIABLE ALONG A CONTINUOUSLY

DIFFERENTIABLE ARC

Definition 7 .3 The definite integral of a continuous complex function
f(z) along (or over) a continuously differentiable arc 1: z = z( t), a :St :S (3,
is defined by the formula

J f(z) dz= L{j J(z(t))z'(t) dt (7.3-1)
'Y
We note that the composite function f(z(t)), as well as z'(t), are con-
tinuous functions of the real variable t on the interval [a, (3]. Hence the
right-hand side of (7.3-1) is an integral of the type considered in Section
7 .2. Thus our definition reduces the integral of a continuous complex func-
tion along a continuously differentiable arc to the integral of a continuous