Cauchy's integral formula

Here is the near-magical fact at the heart of complex analysis: for a holomorphic function, the value at any interior point is completely fixed by its values on a loop around that point. Know f on the boundary and you know f everywhere inside -- the interior carries no independent information. Cauchy's integral formula is the exact statement of this 'the boundary knows all' rigidity.

If f is holomorphic on and inside a simple closed contour C, then for any point a inside, f(a) = (1/(2 pi i)) times the contour integral over C of f(z)/(z - a) dz. Differentiating under the integral gives the value of every derivative from the same boundary data: f^(n)(a) = (n!/(2 pi i)) times the contour integral of f(z)/(z - a)^(n+1) dz. This is really the residue theorem applied to the integrand f(z)/(z - a), whose pole at z = a has residue f(a).

It is the source of complex analysis's rigidity theorems: because all derivatives exist, a holomorphic function is automatically analytic (equal to its own Taylor series), and Liouville's theorem (a bounded entire function is constant) and the maximum-modulus principle follow at once. In physics the same 'boundary determines interior' logic is exactly what makes dispersion relations work: causality forces a response function to be holomorphic in a half-plane, and its real and imaginary parts (refraction and absorption) are then locked together by the Kramers-Kronig relations.