path independence
Imagine two hikers leaving the same trailhead for the same summit by completely different trails. For most journeys the effort spent depends on the route. But for some special quantities — height gained, say — only the start and end matter. Path independence is the statement that a contour integral behaves like height gained: the value of the integral of f between two points is the same no matter which contour you take, as long as you stay in the relevant domain.
Why would an integral have this property? Because it is equivalent to two other things, and they all stand or fall together. On a connected open set, the following are the same property of a continuous f: (1) the integral of f between any two points is path-independent; (2) f has a primitive on the set; (3) the integral of f around every closed contour in the set is zero. From (2) the fundamental theorem gives (1) and (3) immediately; conversely, if loops vanish you can define a primitive by F(z) = integral of f from a fixed base point to z, the path-independence guaranteeing this is well-defined.
This three-way equivalence is the conceptual engine of the chapter. Cauchy's theorem will supply path-independence for holomorphic functions on simply connected domains for free, and then everything good follows. The honest caveat: independence holds within a domain, and the domain's shape is everything. The same f = 1/z is path-independent on the right half-plane (it has a primitive there, a branch of log z) but not on a ring around the origin, where two routes on opposite sides of the hole disagree by 2 pi i.
Integrate f(z) = 2z from 0 to 2. A straight path and a semicircular detour give the same answer, because 2z has primitive z^2: both equal 2^2 - 0^2 = 4. By contrast 1/z, integrated from 1 to -1 over the upper unit semicircle versus the lower one, gives i pi versus -i pi — path-dependent, because of the hole at 0.
Path-independent (2z, with a primitive) versus path-dependent (1/z around a hole).
Path independence is a statement about a specific domain: enlarging or puncturing the domain can destroy it, so always ask 'independent on what region?' before invoking it.