Contour deformation
Sliding a contour to a more convenient one without changing a contour integral's value, as long as no singularity is crossed.
Contour deformation is the principle that a contour integral of an analytic function is unchanged when the contour is continuously slid to another contour with the same endpoints (or, for a closed contour, to another closed contour), provided the region swept out during the slide stays inside the domain where the function is analytic -- no singularity is ever crossed along the way. It follows from Cauchy-Goursat applied to the region between the two contours, and it is the standard justification for replacing an awkward contour with a simple circle or line around the same singularities.
Examples
A large circle around the origin can be deformed down to a small circle around the origin without changing \oint 1/z\,dz, since 1/z is analytic on the whole annulus between them.
Two contours from z_0 to z_1 that together bound a region free of singularities give the same contour integral of any function analytic on that region.