holomorphic, analytic and regular
Reading complex analysis you will meet three words used for the same class of functions: holomorphic, analytic, and (in older or applied texts) regular. In this subject they all mean the same thing, but they START from different definitions, and that they coincide is a genuine theorem, not a convention.
'Holomorphic' is the differential definition: f is complex-differentiable on an open set. 'Analytic' is the series definition: around every point f equals a convergent power series sum a_n (z - z_0)^n. These sound very different — one is about a limit of difference quotients, the other about an infinite sum — yet a major result (proved later via Cauchy's integral formula) shows that on an open set the two are EXACTLY equivalent. 'Regular' is an old synonym, common in physics and engineering, meaning the same complex-differentiable property.
Why keep both words if they mean the same thing? Because each viewpoint pays off differently. The holomorphic viewpoint makes the Cauchy-Riemann equations and conformality natural; the analytic viewpoint makes Taylor coefficients, zeros, and analytic continuation natural. Beware that in REAL analysis 'analytic' (equals its Taylor series) is strictly stronger than 'differentiable' (a real smooth function need not be analytic) — the equivalence is a special gift of the complex world.
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 is identically 0). In the complex world this gap closes: holomorphic on an open set forces analytic, and vice versa.
The holomorphic = analytic equivalence is a complex-analysis theorem with no real-variable counterpart.
Treat the three words as interchangeable here, but remember the equivalence holomorphic = analytic is proved, not assumed — and it is precisely what fails in real analysis.