real and imaginary parts of contour integral

If $f(z)$ is continuous on the contour $\gamma$ of the complex plane and

$$z=x+iy\mathit{\hspace{1em}}(x,y\in ℝ),\mathrm{\Re }f(z)=u(x,y),\mathrm{\Im }f(z)=v(x,y),$$

then the contour integral of $f(z)$ along $\gamma$ may be expressed via two real path integrals as

$${\int }_{\gamma }f(z)𝑑z={\int }_{\gamma }(udx-vdy)+i{\int }_{\gamma }(vdx+udy).$$