# real and imaginary parts of contour integral

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

 $z\;=\;x\!+\!iy\quad(x,\,y\in\mathbb{R}),\qquad\Re{f(z)}\;=\;u(x,\,y),\qquad\Im% {f(z)}\;=\;v(x,\,y),$

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

 $\int_{\gamma}f(z)\,dz\,\;=\;\int_{\gamma}(u\,dx-v\,dy)+i\!\int_{\gamma}(v\,dx+% u\,dy).$
Title real and imaginary parts of contour integral RealAndImaginaryPartsOfContourIntegral 2013-03-22 19:14:32 2013-03-22 19:14:32 pahio (2872) pahio (2872) 6 pahio (2872) Result msc 26B20 msc 30E20 msc 30A99 ContourIntegral PathIntegral RealPartSeriesAndImaginaryPartSeries