A zero, and how hard the function presses on it
In the previous guide you cashed in Taylor's theorem: a holomorphic function is analytic, equal near each point z_0 to a convergent power series sum a_n (z - z_0)^n with a_n = f^(n)(z_0)/n!. We now turn that series loose on one simple question — where does f equal zero? A zero of f is just a point where f(z_0) = 0. For a polynomial these are the familiar roots; for a general analytic function they turn out to be far more structured, and that structure is the seed of nearly everything rigid in the subject.
Here is the one fact to hold onto. If f is analytic near z_0 and f(z_0) = 0 but f is not identically zero, then the Taylor series has lost its constant term, and the first surviving coefficient sits at some index m: f(z) = a_m (z - z_0)^m + a_(m+1)(z - z_0)^(m+1) + ..., with a_m not 0. Pull out the lowest power and you get f(z) = (z - z_0)^m g(z), where g is analytic and — crucially — g(z_0) = a_m is not zero. That single integer m is the order of the zero, the analytic mirror of the multiplicity of a polynomial root.
f analytic near z_0, f(z_0) = 0, f not identically 0
Taylor series: f(z) = a_m (z - z_0)^m + a_(m+1)(z - z_0)^(m+1) + ...
with a_m =/= 0 (m = order of the zero)
factor out: f(z) = (z - z_0)^m * g(z)
g analytic, g(z_0) = a_m =/= 0
equivalently: f(z_0) = f'(z_0) = ... = f^(m-1)(z_0) = 0, f^(m)(z_0) =/= 0Read m as how hard the function presses on zero. Order 1 — a simple zero — is a clean crossing, like z passing through the origin. Order 2 is a tangency: the function flattens against zero, with f and f' both vanishing, like 1 - cos z = z^2/2 - z^4/24 + ... starting at z^2. A neat sanity check from the Taylor coefficients: order m means f(z_0) = f'(z_0) = ... = f^(m-1)(z_0) = 0 while f^(m)(z_0) is not — count the derivatives that die before one survives, and you have the order.
Why every zero stands alone
Now squeeze the factorization for its first big consequence. Near a zero z_0 of order m we have f(z) = (z - z_0)^m g(z) with g(z_0) not zero. Since g is continuous and nonzero at z_0, it stays nonzero on some small disk around z_0 — continuity does not let it suddenly hit zero. On that disk the only way f can vanish is through the factor (z - z_0)^m, and that is zero only at z_0 itself. So z_0 is the lone zero in its little disk. This is the isolation of zeros: a nonzero analytic function's zeros never crowd together, never fill a segment, never pile up — each gets breathing room.
The word nonzero in that sentence is not decoration. If f were identically zero it would vanish everywhere, and isolation would collapse trivially — every point a zero, none of them alone. Isolation is a statement about functions that are not the zero function; for any such f, the factor-out argument applies at each of its zeros. Even an infinite zero set stays well-behaved: sin z is entire and vanishes exactly at z = n pi for every integer n — infinitely many zeros, yet no two are closer than pi, each crisply isolated.
From isolation to the identity theorem
Isolation hands us the headline result almost for free. The identity theorem asks: how much of an analytic function do you need to know to know all of it? Astonishingly little. If f is analytic on a connected region and equals zero on any set that has a limit point inside the region — even just a single convergent sequence of points — then f is identically zero on the whole region. There is no freedom to be zero on the sequence and nonzero elsewhere; the local data has already fixed the global function.
The proof is isolation in disguise. Suppose the zeros of f have a limit point z* inside the region. If f were not identically zero, every one of its zeros would be isolated — but z* is a zero (by continuity) with other zeros arbitrarily close to it, so z* is a non-isolated zero. Contradiction. The only escape is that f vanishes identically near z*, and a connectedness argument then spreads that 'identically zero' across the entire region, one overlapping disk at a time.
- Given two analytic functions f and g that agree on a set with a limit point inside the connected region, form the difference h = f - g.
- h is analytic and vanishes on that set, so the zeros of h have a limit point inside the region.
- Non-isolated zeros are impossible for a nonzero analytic function, so h must be identically zero — that is, f = g throughout the region.
The rigidity you just unlocked
Step back and feel how strong this is. A holomorphic function has no local wiggle room: you cannot nudge it on a tiny patch and keep it analytic, because the patch and the rest would agree on a set with limit points and so would have to be the same function. This is the rigidity that makes complex analysis feel almost magical — and it is the exact opposite of the floppiness of smooth real functions, which you can bend locally at will.
One immediate payoff: a real function like sin x or e^x has at most one analytic extension to the complex plane. The real axis has limit points, so any two analytic functions matching sin x there must agree everywhere — the complex sine is forced, not chosen. This same logic underwrites analytic continuation: if you can push a function past its original domain at all, the extension is unique, because any two extensions agree on the overlap and hence, by the identity theorem, everywhere they both live.
And it makes real identities portable for free. The permanence of functional relations says an analytic identity true on the real line is automatically true on the whole plane: e^(z+w) = e^z e^w and sin^2 z + cos^2 z = 1 are first proved for real arguments, then carried to all complex z because both sides are analytic and agree on a set with limit points. One honest caveat, though — both sides must genuinely be analytic. Relations involving |z|, the conjugate z-bar, or a branch choice are not analytic and need not extend; |e^z| = e^x, for instance, does not lift to a complex identity.
Reading the fine print honestly
A few honest cautions keep these results from being misremembered. First, the order of a zero is always a finite positive integer for a nonzero analytic function — never a fraction, never infinite. An 'infinite-order zero' would mean every Taylor coefficient vanishes, which is just the identity theorem telling you the function is identically zero on its disk. Second, isolation can genuinely fail at the boundary of the domain or at a singularity: sin(1/z) has zeros at 1/(n pi) piling up toward 0, but 0 is an essential singularity, not an interior point of holomorphy, so there is no contradiction — the hypothesis 'analytic at the limit point' is exactly what is missing there.
Third, do not over-read the identity theorem as 'two functions equal at a few points must be equal.' Equality at finitely many points says nothing; equality at the integers says nothing; you need an accumulation point inside the connected region. The theorem is powerful precisely because its hypothesis is so cheap — a single converging sequence — but that hypothesis is genuinely required, not a formality to be skipped.
Carry one picture out of this guide: a nonzero analytic function vanishes only on a sparse scatter of isolated dots, each marked with a finite order m read straight off the first surviving Taylor coefficient. That scatter is the function's fingerprint — and the identity theorem says the fingerprint, sampled along even one converging sequence, already determines the whole function. The next guide turns this same series-and-zeros viewpoint outward, asking how far a Taylor series reaches before it hits the nearest place the function misbehaves.