Taylor's theorem
/ TAY-lor /
Taylor's theorem in complex analysis is the climax of this whole part of the subject, and it is far stronger than its real-variable cousin. It says: if f is holomorphic on a disk centred at z_0, then on that entire disk f equals its Taylor series. Not approximately, not up to a small remainder you have to bound — exactly equal, as an honest convergent power series. Every holomorphic function, near any point of its domain, simply IS a power series.
Precisely, suppose f is holomorphic on the disk |z - z_0| < R. Then for every z in that disk, f(z) = sum_{n>=0} a_n (z - z_0)^n with coefficients a_n = f^(n)(z_0) / n!, and the series converges throughout the disk. The proof runs through the Cauchy integral formula: write f(z) using the integral over a circle around z_0, expand the Cauchy kernel 1/(w - z) as a geometric series in (z - z_0)/(w - z_0), and integrate term by term (justified by uniform convergence). The integral of each term produces exactly f^(n)(z_0)/n! by the generalized Cauchy formula. So the coefficients come from contour integrals, which is why the theorem works in the complex setting and not the real one.
Contrast with calculus on the line, and the miracle stands out. There, a function can be infinitely differentiable yet NOT equal its Taylor series — the famous e^(-1/x^2) extended by 0 has every derivative zero at the origin, so its Taylor series is identically 0 while the function is not. No such pathology can occur for holomorphic functions: one complex derivative on a region forces the function to be analytic, equal to its Taylor series on every disk that fits inside the region. This is the deepest reason the words 'holomorphic' and 'analytic' mean the same thing.
f(z) = 1/(1 + z^2) is holomorphic at z_0 = 0 with nearest singularities at z = +i and z = -i, distance 1. Taylor's theorem promises a series valid on |z| < 1; indeed 1/(1 + z^2) = 1 - z^2 + z^4 - z^6 + ... On the real line this looks mysteriously limited to |x| < 1 despite f being smooth everywhere — the complex singularities at +-i explain the radius.
A function smooth on all of the real line can still have a finite Taylor radius, set by invisible singularities off the real axis.
The series converges only inside the largest disk free of singularities, not on the whole domain of f. Taylor's theorem is a LOCAL statement: a single power series rarely represents a holomorphic function everywhere it is defined.