Suppose $f$ is continuously complex differentiable on $\Omega$ , and $T \subset \Omega$ is a triangle whose interior is also contained in $\Omega$ . Apply Green's theorem to show that $$\int_T f(z) \, dz=0$$
This is an exercise in Stein's complex analysis Page$65$ .
Let $f(z)=u(x,y)+iv(x,y)$ , $dz=dx+idy$ then apply Green's theorem I can get the desire conclusion . But I can not prove that $dz=dx+idy$ , since the definition of integral along a curve only has one variable . $$\int_T f(z) \, dz=\int_a^b f(g(t)) g'(t) \, dt$$