1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 419

where tk-1 < rf. < tk and tk-1 < rf < tk. Hence

n n

L f(z(rk))[z(tk)-z(tk-1)] = L f(z(rk))x'(rf.)(tk -tk-1)

k=l k=l

n

+ i L f(z( Tk))y'( rf.')(tk - tk-1)

k=l

As IPI -+ 0 the Duhamel sums on the right tend, respectively, to

LP f(z(t))x'(t)dt and LP f(z(t))y'(t) dt

Thus we obtain

n 1p

lim L f(z( rk))[z(tk) - z(tk-1)] = f(z(t))[x'(t) + iy'(t)] dt

JPl->O k=l OI '

= 1: f(z(t))z'(t) dt

We note that if f(z) reduces to the real function f(x) of the real variable
x, and if 'Y is the real interval [a, b], we have

n b

J

f(z)dz= lim Lf(xk)(xk-Xk-1)= { f(x)dx

~ JPJ->O k=l. la

so that the complex integral reduces to the usual Riemann definite integral

of f over [a, b]. ·

Also, if f(z) is a continuous complex function of the real variable x,
namely, f(z) = u(x) + iv(x), over [a,b], we have

J

f(z) dz= lim I)u(xi;) + iv(xk)](xk - Xk-1)

~ IPl->O k =1

n n

= lim L u(xk)(xk - Xk-1) + i lim L v(xk)(xk - Xk-1)

IPJ->0 k=l IPJ->0 k=l

= lb u(x) dx + i lb v(x) dx

Hence (7.6-2) contains (7.2-1) as a special case.
For later usage the following definition is introduced: