1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 411

also holds for k complex, let k = k 1 + ik 2 (k 2 I= 0). Then we have

k lb f(t) dt = (k1 + ik 2 ) [lb u(t) dt + i lb v(t) dt]

~ [k, J.' udt-k, J.' vdt] ++ J.' udt+k, J.' vdt]

= 1b(k1u-k2v)dt+i 1b(k2u+k1v)dt

= lb kf(t) dt

Property 6 follows at once from (7.2-1)

Property 7 is trivially true if l: f(t) dt = 0. If l: f(t) dt I= 0, let

lb f(t) dt = rei^8

where r = I l: f(t) dtl and () = Arg l: f(t) dt. Then we have

r = e-iO lb f(t) dt =lb e-iO J(t) dt =Re lb e-iO f(t) dt

b b b '

= 1 Re[e-i^8 f(t)] dt:::; 11e-i^8 f(t)I dt = 1 lf(t)I dt

where we have made use of properties 1 and 5, the fact that r is real, and
the inequality Rew :::; jwj. Clearly, equality holds iff

Re[e-i^8 f(t)] = lf(t)I

that is, iff e-iB f(t) ;'.'.: 0. Thus equality holds iff, for some n,

arg f(t) + 2mr =; () = Arg lb f(t) dt

whenever J(t) I= 0.
To prove 8 it suffices to note that the assumptions imply that g( ,,-) is
a continuous function on [c, d] and that g([c, d]) is also an interval [A, B]
(a compact and connected set) that contains the interval [a, b] (Fig. 7.1).
Hence by a known property of real integrals (see [2], p. 216),

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

and