From counting to a local portrait
Across this rung you built a machine that counts. The argument principle turned an integral of f'/f into a number of zeros minus poles; the winding number of the image curve gave you that count geometrically; Rouche's theorem let you locate zeros by comparison; and the open mapping theorem told you that a non-constant holomorphic map sends open sets to open sets. This last guide cashes all of that in at once. We stop asking how many zeros lie in a big region and instead stand at a single point and ask: what does the map actually look like up close?
Here is the punchline, stated before we earn it. Near a point z_0 a holomorphic map has exactly two behaviours, and which one you get is decided by a single derivative. If f'(z_0) is not zero, the map is locally a clean one-to-one distortion: it spins and scales but never folds, and it has a holomorphic inverse. If f'(z_0) = 0 — a critical point — then the map folds, wrapping a small disk some whole number m of times around its image. There is no in-between and no third case. That is an astonishing amount of rigidity, and the counting tools are exactly what prove it.
The local mapping theorem: every value is hit m times
Let w_0 = f(z_0), and suppose f(z) - w_0 has a zero of order m at z_0. (If f'(z_0) is not zero then m = 1; a critical point is exactly the case m >= 2.) The local mapping theorem says: for every value w close enough to w_0, but w not equal to w_0, the equation f(z) = w has exactly m distinct solutions in a small disk around z_0, each a simple zero. In words, near z_0 the map is m-to-1. It takes the single value w_0 once (with multiplicity m piled on one point) and every nearby value exactly m times, spread over m separate points.
The proof is the argument principle, used with a steady hand. Pick a circle so small that z_0 is the only solution of f(z) = w_0 inside it and f' has no other zero there. The number of solutions of f(z) = w inside that circle is the zero-counting integral of f'(z)/(f(z) - w) around the circle. At w = w_0 that integral equals m. But the integral varies continuously with w while always being a whole number — and a continuous integer-valued function on a connected set is constant. So it stays equal to m for every nearby w. Because f' has no zero except at z_0, those m roots are simple, hence genuinely m distinct points.
f(z) - w_0 ~ a_m (z - z_0)^m (a_m != 0, m >= 1)
solutions of f(z) = w near z_0 : m for every w near w_0, w != w_0
m = 1 <=> f'(z_0) != 0 (locally one-to-one)
m >= 2 <=> f'(z_0) = 0 (critical point, folds m times)Why it forces open-ness and the maximum principle
Look again at the conclusion: every value near w_0 is actually attained. That means the image of a small disk around z_0 contains a whole disk around w_0 — the image has no missing points, no holes, no boundary creeping inward. That is exactly what it means for the map to be open, and so the local mapping theorem is the open mapping theorem seen up close. The local degree m is just the number of preimages, the same m the argument principle counted.
Open-ness instantly gives the maximum principle a transparent reason. Suppose |f| had a local maximum at z_0. The image of a small neighbourhood of z_0 is an open set containing w_0 = f(z_0), so it contains points of strictly larger modulus than |w_0| — there are values f(z) with |f(z)| > |f(z_0)| arbitrarily close by. That contradicts z_0 being a maximum. So a non-constant holomorphic function can never have an interior maximum of its modulus; the maximum modulus principle is forced. No estimate, no integral — just the geometric fact that the image is open and a maximum would need the image to stop at w_0.
The non-critical case: a holomorphic inverse
Now specialise to m = 1, the case f'(z_0) is not zero. Then near z_0 every nearby value is hit exactly once: f is locally one-to-one (injective on a small disk). A one-to-one continuous map onto an open set has an inverse g defined near w_0, with g(w_0) = z_0. The holomorphic inverse function theorem says more: that inverse is itself holomorphic, and its derivative is the reciprocal of f' at the matching point, g'(w) = 1 / f'(g(w)) — the same formula you know from real calculus, now living in the complex plane with full force.
Why is the inverse holomorphic and not merely continuous? Because holomorphy is detectable by an integral. One clean route writes the inverse explicitly: g(w) equals the contour integral of z times f'(z) / (f(z) - w) over a small circle, divided by 2 pi i. That integral is the location of the unique root of f(z) = w — a weighted version of the argument principle — and it depends holomorphically on the parameter w because you can differentiate under the integral sign. So the local inverse is handed to you by the very counting integral that started the rung. This is genuinely stronger than the real inverse function theorem, where a smooth inverse takes a separate continuity-of-derivatives argument.
- Check the test: confirm f'(z_0) is not zero. If it vanishes you are in the folding case and there is no single-valued local inverse — skip to the next section.
- Conclude local one-to-one-ness: by the local mapping theorem with m = 1, f is injective on some small disk D around z_0 and maps D onto an open set around w_0 = f(z_0).
- Name the inverse g and pin its value: g maps that image back to D with g(w_0) = z_0; it exists and is single-valued precisely because f was injective on D.
- Upgrade to holomorphic: g is holomorphic (e.g. via the location integral above), and differentiating f(g(w)) = w gives g'(w) = 1 / f'(g(w)).
The critical case: a clean m-th root model
When f'(z_0) = 0 the picture is not chaos — it is a single, completely understood shape. If the first non-vanishing derivative at z_0 is the m-th, then there is a holomorphic change of coordinate phi, itself invertible near z_0, so that f(z) = w_0 + phi(z)^m on a small disk. Read that slowly: after a smooth holomorphic relabelling of the input, f is literally the m-th power map. All the folding, all the m-to-1 behaviour, is concentrated into the one function z to the m, the cleanest model there is.
Picture the squaring map z to z^2 at the origin, where m = 2. A ray at angle theta goes to a ray at angle 2 theta, so the half-disk above the real axis already covers the whole punctured disk once, and the lower half covers it a second time: the small disk wraps twice around the image, and every non-zero value near 0 has two square-root preimages. A general critical point of order m is this exact movie with the dial set to m. This is also why no single-valued inverse exists there: inverting z^m means choosing an m-th root, which forces a branch point and a branch cut — the multivaluedness is unavoidable, not a failure of effort.
The honest big picture
Step back and the whole rung snaps into one frame. The integral of f'/f counts; the change in argument of the image makes that count visible; Rouche's theorem and Hurwitz's theorem let zeros move around without secretly appearing or vanishing; the open mapping theorem turns the count into geometry; and the local mapping theorem reads off, point by point, the only two shapes a holomorphic map is allowed: an invertible twist where f' is non-zero, an m-fold power fold where it vanishes. Counting was never the goal. It was the lever that pried open the local structure of holomorphic maps.
Two honest caveats before you climb on. First, everything here is local: it describes a small disk and says nothing global. f'(z) never zero gives a local inverse at each point, but as e^z shows, no global inverse need exist; gluing local inverses into one is a question about branches, monodromy, and the Riemann surface, answered on later rungs, not here. Second, the m-th power normal form is exact and beautiful but it lives only near the point — push the disk outward and other critical points and the function's global shape intrude. Local structure is a microscope, not a map of the country; knowing the microscope perfectly is exactly what lets you read the larger map later.