上式的另一表示形式 因为曲线C的参数方程 z(t)=x(t)+iy(t)(a≤t≤B 根据线积分的计算方法,所以 f(adz u(r, y)dx-u(a, y)dy +i /v(, y)dz +u(z, y)dy {ulz(t),y(t)]x'(t)-oz(t),y(t)]y(t)}dt +iu[(t), y(t)r(t)+uz(t), y(t)ly'(t))dt 上式右端可以写成 Tulc(t), y(t)]+iu[r(t),y(t)1) a'(t)+iy(t))dt f(2(t)2(t)dt• þª,L«/ª ÏǑ­‚ C ëꐧ z(t) = x(t) + iy(t) (α 6 t 6 β), Šâ‚È©OŽ{, ¤± Z C f(z)dz = Z C u(x, y)dx − v(x, y)dy + i Z C v(x, y)dx + u(x, y)dy = Z β α {u[x(t), y(t)]x ′ (t) − v[x(t), y(t)]y ′ (t)}dt +i Z β α {v[x(t), y(t)]x ′ (t) + u[x(t), y(t)]y ′ (t)}dt þªmàŒ±¤ Z β α {u[x(t), y(t)] + iv[x(t), y(t)]} · {x ′ (t) + iy′ (t)}dt = Z β α f[z(t)]z ′ (t)dt. ¤± Z Z β 9/127