Three words, three different-sounding promises
By now you have met the complex derivative and watched the Cauchy-Riemann equations fall out of demanding that one limit exist from every direction. The natural next question is: what do we call a function that has this property not just at one point, but everywhere it lives? The textbooks reach for three words — holomorphic, analytic, and regular — and a beginner reasonably suspects these must be three subtly different things. They are not. Sorting out why is the whole job of this guide.
Start with the literal promise each word makes, because the words come from different starting points. Holomorphic means "complex-differentiable at every point of an open set." The key qualifier is open: we never ask for differentiability at a single isolated point only, but throughout a little disk around each point, so the derivative has room to breathe in every direction. Analytic means something that sounds far stronger: "locally equal to a convergent power series," that is, near each point z_0 the function agrees exactly with some sum a_n (z - z_0)^n. Regular is the old-fashioned third name, common in British texts, and it simply means the same as holomorphic — well-behaved, no singular points lurking.
Holomorphic and regular: same thing, different accent
The easy pair first. Holomorphic and regular are flat-out synonyms — there is no theorem to prove, only a vocabulary fact to absorb. The word "holomorphic" comes from Greek roots meaning roughly "whole-shaped," hinting that the function is built from one undivided piece; "regular" is the plainer English word for the same good behaviour. You will meet "regular" mostly in older or British-influenced books, and "holomorphic" almost everywhere else today. When an author writes one, you may silently read the other.
Both words are about an open set, and this is worth pausing on. Asking that f be complex-differentiable at a single point — say only at z_0 = 0 — is almost meaningless, because the magic of complex analysis comes from the derivative existing on a neighbourhood, so that nearby points can talk to each other. That is exactly why "holomorphic on an open set" is the working notion, and why a function holomorphic on the whole plane earns its own special name: an entire function. Polynomials, e^z, cos z and sin z are all entire; they are differentiable everywhere with no exceptions.
The deep equivalence: one derivative becomes a power series
Now the part that genuinely deserves astonishment. The headline theorem of this whole subject says that holomorphic and analytic are the same thing — being complex-differentiable on an open set, by itself, forces the function to equal a convergent power series near every point. One derivative buys you infinitely many. Stop and feel how outrageous this is on the real line: there a function can be differentiable exactly once, with a derivative so ragged it is not differentiable a second time, and a function can be infinitely differentiable yet still NOT equal its own Taylor series. Complex differentiability slams all of those doors at once.
Where does this gift come from? You will see the full machinery in a later rung, but the engine is the contour integral: from the integral of f around a small loop you can read off, in one stroke, the value of f and of every one of its derivatives at the centre. Concretely, the Cauchy integral formula recovers f(z_0) from the boundary, and the same idea differentiated under the integral sign delivers f'(z_0), f''(z_0), and so on without ever stopping.
integral of f(z)/(z - z_0) dz = 2 pi i f(z_0)
once around a small loop ...
... gives the value at the centre,
and differentiating under the integral
gives f'(z_0), f''(z_0), f'''(z_0), ... forever.
=> f(z) = sum a_n (z - z_0)^n near z_0
with a_n = f^(n)(z_0) / n!The upshot is a property called infinite differentiability: a holomorphic function is automatically smooth to every order, and more — it is equal to its Taylor series on a disk, not merely tangent to it. This is the single most important way complex analysis departs from real calculus, and it is why we are so careful with the word "analytic." In a complex course, calling a function analytic is not a stronger claim than calling it holomorphic; it is the very same claim, viewed through the power-series lens instead of the derivative lens.
Reading the labels honestly: the small print
Three honest cautions keep these words from misleading you. First, the equivalence "holomorphic equals analytic" is special to the complex setting; do not carry it back to the real line, where the words come apart badly. Second, in the previous guide you saw that the Cauchy-Riemann equations alone are not quite enough to guarantee holomorphy — you need the partial derivatives continuous (or f real-differentiable as a map of two variables) so that the sufficient condition applies. A function can satisfy u_x = v_y and u_y = -v_x at a stray point and still fail to be differentiable there.
Third, watch the boundaries of the region. "Holomorphic on a set" should be read as holomorphic on an open set, with no claim made about the edge. And many of the friendliest functions are holomorphic only after you remove some bad points: 1/z is holomorphic on the whole plane except the origin, where it has a singular point. Log z is holomorphic only once you cut a branch and slit the plane open. Naming a function "holomorphic" without naming WHERE is half a sentence.
Why this single idea earns five guides
If holomorphic, analytic, and regular all name one idea, why does that idea deserve a whole rung? Because almost every striking theorem in complex analysis is really a sentence that begins "if f is holomorphic, then ..." and ends somewhere astonishing. A holomorphic function is pinned down by a tiny sample of itself; it cannot have an interior maximum of its modulus; its real and imaginary parts are quietly solving Laplace's equation. None of these follow from real differentiability — they all flow from the rigidity we just uncovered.
- Start with one fact: f is complex-differentiable on an open set (holomorphic).
- The same fact, restated through power series, makes f analytic — equal to sum a_n (z - z_0)^n near every point.
- Now infinite differentiability, the identity theorem, the maximum modulus principle, and harmonic real parts all become consequences, not new assumptions.
So when you meet "holomorphic," "analytic," or "regular" in the chapters ahead, do not waste energy on telling them apart — they are one idea wearing three hats. Spend that energy instead on the thing they all promise: a single complex derivative, asked of an open set, is a contract so strong that it hands you a power series, infinite smoothness, and the entire cascade of complex analysis in return. The next two guides cash in two pieces of that contract — the derivative seen as a local rotation-and-scaling, and the harmonic partners hiding inside u and v.