determination by boundary values
Here is the philosophical punchline of Cauchy's formula stated on its own: a holomorphic function is so rigid that its values on a closed curve completely pin down its values everywhere inside. There is no freedom left. If two holomorphic functions agree on a surrounding loop, they must agree at every interior point. Knowing the edge is knowing the whole.
Why this is startling: in ordinary real calculus a function on a disk can do almost anything inside while still matching some given boundary values — there are infinitely many smooth functions with the same boundary data. Holomorphy removes that freedom. Because f(z_0) = (1 / (2 pi i)) times the integral over gamma of f(z) / (z - z_0) dz depends only on the values f(z) along gamma, fixing those boundary values fixes f(z_0) by a definite formula, point by point. The interior is not a separate stage to fill in; it is forced.
This rigidity has deep relatives. The maximum-modulus principle and the identity theorem are other faces of the same stiffness: a holomorphic function cannot vary independently in different parts of its domain. The same idea drives the Dirichlet problem for harmonic functions, where prescribing values on a boundary determines a unique harmonic function inside. The lesson worth carrying: holomorphic functions are global objects in disguise — a local foothold, or a boundary, controls the rest.
Two entire functions whose values coincide along the unit circle must coincide throughout the open disk; you cannot bend one of them inside without breaking holomorphy.
Agreement on a loop forces agreement inside — there is no interior wiggle room.
This is determination by boundary values on a closed curve inside the domain of holomorphy, not by values on the boundary of the domain itself, which can be subtler (boundary behavior may be wild even when the interior is tame).