Cauchy's integral formula
Cauchy's integral formula is one of the most astonishing statements in mathematics: for a holomorphic function, the values on the boundary of a region completely determine every value inside. Knowing f only on a circle, you can reconstruct f at the center, and at every other interior point, by a single integral. The interior is held rigidly hostage to the boundary — there is no freedom to wiggle the inside while keeping the edge fixed.
Stated precisely, if f is holomorphic on and inside a positively-oriented simple closed contour gamma, then for any point a inside, f(a) equals (1 / (2pi i)) times the contour integral over gamma of f(z) / (z - a) dz. Differentiating under the integral sign yields formulas for every derivative: f^{(n)}(a) equals (n! / (2pi i)) times the contour integral of f(z) / (z - a)^{n+1} dz. This single formula instantly proves that a holomorphic function is infinitely differentiable.
The consequences cascade. The formula yields Cauchy's estimates bounding the derivatives by the size of f on the boundary, from which Liouville's theorem follows in two lines. It gives the mean value property: f at the center of a disc is the average of its boundary values. And it is the route by which holomorphic functions are shown to be analytic — expand 1/(z - a) as a geometric series and integrate term by term to get the Taylor coefficients.
To evaluate the integral of e^z / (z - 1) around a circle of radius 2 centered at 0: here f(z) = e^z is holomorphic and a = 1 lies inside, so the formula gives the integral = 2pi i times f(1) = 2pi i times e. The boundary integral has pinned down the value e^1 at the interior point.
Boundary data recovers an interior value.