Physical / geometric interpretation of complex integration
There are many geometric and physical interpretations of integrals of real-valued functions (such as are, volume, work etc.). However, a similar interpretation is rarely discussed for complex contour integral. The following interpretation of contour integration is due to Pólya (George Pólya and Gordon Latta, Complex Variables, Wiley, 1974.)
Let [math]\displaystyle{ \gamma }[/math] be a simple closed smooth curve in the complex plane, and [math]\displaystyle{ f }[/math] be a (complex-valued) function defined on [math]\displaystyle{ \gamma }[/math]. Parametrize [math]\displaystyle{ \gamma }[/math] with arc length [math]\displaystyle{ s }[/math]. As usual, have [math]\displaystyle{ f(z) = u(z) + iv(z) = u(x,y) + iv(x,y) }[/math]. Consider a vector field [math]\displaystyle{ {\bf F} (x,y) = [u(x,y), -v(x,y)] }[/math]