1550251515-Classical_Complex_Analysis__Gonzalez_

434 Chapter^7

If Po - {a = t 0 , t 1 , ••• , in = ,8} is a partition of the interval [a, ,8] and

tk-1 $ 'Tk $ tk, !Pol = max(tk - tk-1), we set

J

P(x,y)dx = lim tP(x(rk),y(rk))[x(tk)-x(tk-1)]

'Y IPo 1-+0 k=l

(7.8-15)

j Q(x,y)dy = lim tQ(x(rk),y(rk))[y(tk)-y(tk-1)]

'Y IPol-+O k=l

(7.8-16)

provided that the indicated limits exist. The limits do exist if P(x, y) and
Q(x, y) are continuous along 'Y and the arc 'Y is rectifiable, in which case
the right sides of (7.8-15) and (7.8-16) become Riemann-Stieltjes integrals
over [a, ,8], so that

j P(x, y) dx = 1: P(x(t), y(t)) dx(t) (7.8-17)

I Q(x,y)dy = 1: Q(x(t),y(t))dy(t) (7 .8-18)

In particular, if 'Y is piecewise continuously differentiable, the right-hand
sides become ordinary Riemann integrals:

J P(x,y)dx = 1: P(x(t),y(t))x'(t)dt

'Y

J Q(x, y) dy = ifi Q(x(t), y(t))y'(t) dt

'Y

Furthermore, we define

IP dx + Q dy = JP dx + J Q dy

(7.9-19)

(7.8-20)

(7.8-21)

Now, to prove (7.8-14), it suffices to apply Definition 7.6, separate
into real and imaginary components and use (7.8-15) to (7.8-18), as well
as (7.8-21). Thus

J

f(z) dz= lim ~ f(z(rk))[z(tk) - z(tk-1)]

IPl-okLt

'Y =1

n

= lim '°'[u(x(rk),y(rk))+iv(x(rk),y(rk))]

IPl-+O~