homotopy invariance of contour integrals
/ homotopy: HOH-muh-toh-pee /
Homotopy is the topologist's word for 'continuous deformation'. Two paths are homotopic (in a region) if one can be continuously slid into the other while staying in the region, keeping their endpoints fixed. Homotopy invariance is the clean theorem that the contour integral of a holomorphic function depends only on the homotopy class of the path: deform the path however you like inside the holomorphic region, and the integral never changes.
This is the deformation principle stated in its proper topological language, and it is the natural home of Cauchy's theorem. If a closed loop is homotopic to a constant point (null-homotopic) inside the region where f is holomorphic, then its integral equals the integral over a point, which is zero — that is exactly Cauchy's theorem. On a simply connected domain every loop is null-homotopic, so every loop integral vanishes; on a domain with holes, two loops give the same integral precisely when they are homotopic, that is, when they wind around the holes the same way.
The practical consequence is that you classify contours by how they thread around the singularities, not by their geometry. Two wildly different-looking loops that encircle exactly the same punctures the same number of times are homotopic in the punctured region and hence give identical integrals. This is why, in residue calculus, you are free to choose the most convenient contour in a homotopy class — a big circle, a rectangle, a keyhole — knowing the answer is fixed by topology, not by the particular curve you drew.
In the plane minus the origin, the upper-half-circle path and the lower-half-circle path from 1 to -1 are not homotopic (one passes above the hole, the other below), so the integral of 1/z along them differs — by exactly 2 pi i, the cost of the loop they jointly form winding once around 0.
Non-homotopic paths around a hole give different integrals; homotopic ones always agree.
Homotopy invariance requires f to be holomorphic throughout the swept region; if a singularity lies between the two paths they need not be homotopic there, and their integrals can legitimately differ.