On the other hand, the definite integral of a complex function f(z)

$\displaystyle

I = \int_C f(z) dz,

$

where C is a smooth curve on the complex plane, can be reduced to real curvilinear integrals as follows:

$\displaystyle

I = \int_C(u dx - v dy) + i \int_C(u dy + v dx),

$

where $\displaystyle z=x+iy, f(z)=u(x,y)+iv(x,y)$. Then, it seems to me that it is possible to define the definite integral even if f(z) has, at least, a finite number of discontinuities on C as in the integrals of real functions.