the equivalence of analytic and holomorphic
Two words float around the same idea and it is natural to wonder whether they differ. Holomorphic means complex-differentiable: f'(z) exists at every point of a region. Analytic means locally representable by a convergent power series: around every point, f equals some sum a_n (z - z_0)^n on a small disk. In real analysis these are genuinely different properties. The great theorem of complex analysis is that here they are exactly the same: a function on an open set is holomorphic if and only if it is analytic. You may use the two words interchangeably forever.
Why are they equivalent? One direction is easy: a convergent power series can be differentiated term by term inside its disk, so an analytic function is automatically holomorphic. The hard, beautiful direction is the reverse, and it is precisely Taylor's theorem driven by the Cauchy integral formula: merely assuming a single complex derivative exists on a region forces the function to equal its Taylor series on every disk inside. So 'has one complex derivative' secretly means 'has all derivatives and equals its power series'. The mild-sounding hypothesis of complex differentiability is enormously powerful.
This equivalence is the philosophical heart of the whole subject and has no real-variable analogue. On the line, differentiable does not imply twice differentiable, twice differentiable does not imply smooth, and even smooth does not imply analytic. In the complex world the entire ladder collapses to a single rung: one derivative gives you everything. It is why complex analysis feels so rigid and rewarding — almost every local question has a clean answer, because every holomorphic function is, up close, just a power series.
On the real line, g(x) = e^(-1/x^2) (with g(0) = 0) is infinitely differentiable but not analytic at 0 — its Taylor series there is identically 0 yet g is not. There is no complex function with this defect: any f holomorphic near a point necessarily equals its Taylor series on a disk, so analytic and holomorphic coincide.
The real counterexample e^(-1/x^2) has no complex twin — in the plane, one derivative already forces full analyticity.
Some authors define 'holomorphic' to mean complex-differentiable and 'analytic' to mean power-series-representable, then prove they coincide; others take the equivalence for granted and treat the words as synonyms from the start. Either way, in complex analysis they describe the same functions.