If $C$ is a contour in the complex plane, is there a geometric interpretation of the contour integral $\int_Cf(z)dz$? What does the value of this integral mean/tell us about anything? Why does the sign of the integral change when the contour is traversed in the opposite direction?

  • $\begingroup$ Also see Geometric interpretation of complex path integral and Gluchoff's note Simple Interpretation of the Complex Contour Integral in AMM. $\endgroup$ – Conifold Apr 23 '17 at 18:54
  • $\begingroup$ Ah, I was looking for another question on here and I guess just wasn't searching the right terms thanks. One thing that answer didn't address that I don't understand is why the sign of the integral changes when the contour is traversed in reverse. $\endgroup$ – Tyler Apr 23 '17 at 19:14
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    $\begingroup$ This is common to all vector field integrals, not just complex ones. It is a useful convention to make sure that when you traverse the same path one way and then back the end result is 0. It is true even for integrals on the real line, switching the limits of integration is interpreted as reversing the integral's sign, see Why does an integral change signs when flipping the boundaries? $\endgroup$ – Conifold Apr 23 '17 at 19:23
  • $\begingroup$ I never made that connection with the real line case. Thanks! $\endgroup$ – Tyler Apr 23 '17 at 19:30