Cauchy's integral formula
/ koh-SHEE /
Imagine you can only walk along the fence around a field, never stepping inside. Cauchy's integral formula says something that sounds like magic: if a function is holomorphic (complex-differentiable) on and inside that fence, then by recording its values all the way around the fence you can recover its value at any single point inside — exactly, with no extra information. The boundary already knows everything about the interior.
Concretely: let f be holomorphic on a region containing a simple closed contour gamma (traversed once counterclockwise) and its interior, and let z_0 be a point inside gamma. Then f(z_0) = (1 / (2 pi i)) times the integral over gamma of f(z) / (z - z_0) dz. To see where the 2 pi i comes from, take f(z) = 1 and let gamma be the circle |z - z_0| = r. Parametrize z = z_0 + r e^(i theta), so dz = i r e^(i theta) dtheta, and the integrand 1 / (z - z_0) dz becomes i dtheta; integrating theta from 0 to 2 pi gives 2 pi i, and dividing by 2 pi i returns 1 = f(z_0). The factor f(z) inside just samples f near z_0 because the kernel 1 / (z - z_0) concentrates all its weight there.
This single formula is the engine room of the whole subject. From it pour out: that holomorphic functions are infinitely differentiable, the Cauchy estimates, Liouville's theorem, the maximum-modulus principle, and the power-series expansion. It rests on Cauchy's integral theorem (that the integral of a holomorphic function around a closed loop is zero) — the formula is what you get by applying that theorem to f(z) / (z - z_0), whose only trouble spot is the simple pole at z_0.
With f(z) = e^z and gamma the unit circle, the integral over gamma of e^z / (z - 0) dz equals 2 pi i times f(0) = 2 pi i times 1 = 2 pi i, so the contour integral is fixed entirely by the value e^0 = 1.
Plugging f(z) = e^z and z_0 = 0 into the formula: the loop integral is read off from one interior value.
The formula needs z_0 strictly inside gamma; if z_0 lies on the contour the integral is improper, and if z_0 is outside, the integrand is holomorphic everywhere inside and Cauchy's theorem makes the integral zero — not f(z_0).