Is the result of a contour integral around a closed loop path unique? I'd think it'd be similar to a definite integral, which is unique.
This came up regarding the proof of uniqueness of a Laurent Series - if the contour integral is unique, then the definition of the Laurent Series can be used to prove its uniqueness.
Edit - as pointed out below, the Laurent Series aspect may be a limiting case, as it's within a specified annulus, so for any of the Cauchy coefficients, the same analytic annulus is used, so the same number of poles will always be within the inner radius.
If you are inegrating a function witth pole(s) around a closed loop, the inegral depends on which pole(s) is/are enclosed by the loop.