Cauchy's integral formula
/ KOH-shee /
Here is one of the most astonishing facts in all of mathematics: for an analytic function, the values along a boundary loop completely determine the value at every point inside. Know f on the rim and you know f throughout the interior, exactly. Cauchy's integral formula is the precise recipe that reconstructs the inside from the boundary, and it is the reason analytic functions are so rigid and so computable.
The formula states that if f is analytic on and inside a positively oriented simple closed contour C, then for any point a inside, f(a) equals 1/(2 pi i) times the contour integral of f(z)/(z - a) dz around C. The integrand has a single pole at z = a, and dividing by z - a is what extracts the value there; encircling that pole multiplies by 2 pi i, which the prefactor undoes. Differentiating under the integral gives a bonus: the same boundary data yields every derivative, f^(n)(a) = n!/(2 pi i) times the integral of f(z)/(z - a)^{n+1} dz. This is why analytic functions are automatically infinitely differentiable — the formula manufactures all their derivatives from boundary values.
The applied consequences are deep. It proves the mean-value and maximum-modulus properties used in potential theory, underlies error bounds for Taylor series, and gives a numerically stable way to compute derivatives by integrating around a circle (avoiding the cancellation that plagues finite differences). It is the special case (a simple pole, residue f(a)) from which the general residue theorem grows, and conceptually it is the seed of holography: complete interior information stored on a lower-dimensional boundary.
To find the integral of e^z/(z - 1) around a circle of radius 2 centred at the origin, recognise it as Cauchy's formula with f(z) = e^z and a = 1 (which sits inside the circle). The answer is 2 pi i times f(1) = 2 pi i times e^1 = 2 pi e i. The boundary integral handed back the interior value e^1 with no further work.
The formula reads off an interior value directly from a boundary integral — no antiderivative needed.
The point a must lie strictly inside the contour, and f must be analytic on the whole enclosed region including a. If a is outside the loop the integral is zero instead (by Cauchy's theorem); if f has its own singularity inside, you are in residue-theorem territory and must account for it.