So I recently took a course that involves contour integration. I understood how to perform the integrals out, but I never got a hold of what the physical meaning was. I understand the introductory meaning of the integral $\int_a^b f(x)\space dx$ as the area under the curve f(x).
However, once I got into complex analysis, we dealt with nasty integrals (such as Cauchy)$$f(a)=\frac 1 {2\pi i} \oint_C \frac {f(z)} {z-a} dz$$I do not quite understand what the meaning of this is. The result of the integral depends on C, just as any integral depends on the bounds. But what is this saying. It is not the area under f(z), but what is it?
