Advanced Calculus fi..

Chapter 8 Functions of a Complex Variable 541Figure 8.3Complex line integral.') f, tobtThe complex integral ff (z) dz is defined as a line integral, and its properties areclosely related to those ,of the integral [P dx + Q d y (see Chapter 5).Let C be apath from A to B in the complex plane: x = x(t), y = y(t), a 5 t .5 b.We assume C to have a direction, usually that of increasing t. We subdivide theinterval a 5 t 5 b into n parts by to = a, tl , . . . , t,, = b. We let z, = x(tj) + iy(tj)and A jz = z, - zj-,, A jt = tj - tj-, . We choose an arbitrary value tj in theinterval 5 t 5 ti and set zy = x(r;) + iy(tf). These quantities are all shown inFig. 8.3. We then writeIf we take real and imaginary parts in (8.21), we findCf (2)dz = lirn x ( u + iu)(Ax + i Ay)that is,The complex line integral is thus simply a combination of two real line integrals.Hence we can apply all the theory of real line integrals. In the following, each path