Let \(\gamma\colon[a,b]\to\CC\) be a piecewise \(C^1\) curve and let \(f\) be a continuous complex-valued function on \(\gamma([a,b])\). The contour integral of \(f\) along \(\gamma\) is \[ \int_{\gamma}f\dz\coloneqq\int_a^b f(\gamma(t))\,\gamma'(t)\dt. \]
Write \(f=u+iv\) with \(u,v\) real-valued and \(z=x+iy\), so that \(dz=dx+i\,dy\). Then \(f\dz=(u+iv)\dx+(-v+iu)\dy\), and the contour integral is the line integral of this complex-valued \(1\)-form: \[ \int_{\gamma}f\dz=\int_\gamma (u+iv)\dx+(-v+iu)\dy. \]