The fine print on "holomorphic maps preserve angles"
The previous guide sold you a clean headline: a holomorphic map preserves angles. But read the receipt carefully and you will find a condition stapled to it. Near a point z_0, the map f behaves to first order like multiplication by its derivative f'(z_0) — the amplitwist, a single rotate-and-scale that turns every tiny vector by arg f'(z_0) and stretches it by |f'(z_0)|. Two curves crossing at z_0 get their tangent vectors turned by the same angle, so the angle between them survives. That whole argument quietly assumed one thing.
It assumed the multiplier is not zero. If f'(z_0) = 0, then |f'(z_0)| = 0 and arg f'(z_0) is undefined — there is no rotation angle to speak of, and the scaling factor crushes every first-order vector to nothing. The leading-order picture goes blank, and with it the proof of angle preservation. A point where the derivative vanishes is exactly the place the theorem refuses to cover. We call it a critical point of the map.
Watch the squaring map double every angle at the origin
The cleanest place to catch this in the act is the squaring map f(z) = z^2, whose derivative is f'(z) = 2z. That derivative is non-zero everywhere except at the origin, so z^2 is conformal on the rest of the plane but has a single critical point at z_0 = 0. To see what it does there, write z in polar form as z = r e^(i theta). Squaring gives z^2 = r^2 e^(i 2 theta): the modulus is squared, and — the crucial part — the argument is doubled.
Now send two rays out from the origin, one along the positive real axis at angle 0 and one at angle theta. They meet at the origin with angle theta between them. After the map, the first ray still points along angle 0 (since 2 times 0 is 0), but the second now points along angle 2 theta. The angle between the images is 2 theta — exactly twice what we started with. The squaring map does not preserve the angle at the origin; it doubles it. A quarter-plane (a 90-degree wedge) is opened into a half-plane (a 180-degree wedge).
z = r e^(i theta) f(z) = z^2 = r^2 e^(i 2 theta) ray at angle 0 --> image at angle 0 ray at angle theta --> image at angle 2 theta angle between rays: theta --> 2 theta (DOUBLED at z = 0) Away from 0, f'(z) = 2z is non-zero, and angles are preserved as usual.
The order of the zero is the angle multiplier
Why exactly double, and not some other factor? Because the squaring map vanishes at the origin to a very specific degree. The honest bookkeeping is the order of the zero of the derivative — or, equivalently, the order to which f(z) - f(z_0) vanishes. For f(z) = z^2 we have f(z) - f(0) = z^2, a zero of order 2. For a general holomorphic f with a critical point at z_0, the local Taylor expansion has its first non-constant term at some power n, so that f(z) - f(z_0) = a_n (z - z_0)^n + (higher), with a_n not zero and n at least 2. That integer n is the verdict.
Near z_0 the map then behaves like w - w_0 = a_n (z - z_0)^n, and raising to the n-th power multiplies arguments by n exactly as squaring multiplied them by 2. So a critical point where f(z) - f(z_0) vanishes to order n multiplies every angle at z_0 by n. The squaring map had n = 2, hence the factor of two. A cubing map z^3 would triple angles; f(z) = z - z^2/2, whose derivative 1 - z vanishes at z = 1 to order one (so f - f(1) vanishes to order two there), doubles angles at z = 1. The rule is uniform and quantitative, not a vague "something breaks."
- Compute f'(z) and find the points where f'(z_0) = 0 — these are the candidate critical points.
- At such a z_0, Taylor-expand: write f(z) - f(z_0) = a_n (z - z_0)^n + (higher order) with a_n not zero.
- Read off the order n (the smallest power that actually appears) — that integer is the angle multiplier at z_0.
- Conclude: a wedge of opening angle alpha at z_0 is mapped to a wedge of opening angle n times alpha.
Why a doubled angle also kills injectivity
Critical points are bad news for a second, closely related reason: the map stops being locally one-to-one there. Stay with z^2. The two points z and -z, distinct unless z = 0, both square to the very same value z^2. So every output near the origin (except 0 itself) has two pre-images that have been folded together. The doubling of angle and the two-to-one folding are the same phenomenon seen from two angles: when you wrap a wedge around to twice its opening, you necessarily lay two input directions on top of every output direction.
This is the geometric heartbeat of the holomorphic inverse function theorem: f has a holomorphic local inverse near z_0 precisely when f'(z_0) is not zero. Where the derivative survives, the amplitwist is an invertible rotate-and-scale, so f is a local bijection and you can undo it analytically. At a critical point the amplitwist collapses, the folding sets in, and no single-valued holomorphic inverse can exist on a full neighbourhood — you would need to choose a branch, exactly the way the square root forces a branch cut. Conformality and local invertibility stand or fall together, and they fall at exactly the critical points.
What this means when you map regions
Critical points are not just a curiosity; they are the practical hazard you steer around whenever you transplant a problem from one region to another. If you want a map that smoothly straightens a corner — say opening a 90-degree wedge into a flat half-plane — you reach for z^2 precisely because it doubles angles at the corner point. Here the critical point is the feature, deliberately parked at the troublesome vertex. The boundary corner is allowed to be non-conformal; the interior, where you actually do analysis, must stay clean.
This is also exactly why the standard maps coming later in the rung are chosen so carefully. A Mobius transformation (z + a) / (c z + d) has derivative (ad - bc) / (c z + d)^2, which is never zero as long as ad - bc is not zero — so a genuine Mobius map has no critical points at all and is conformal everywhere on the sphere. That flawless conformality is the secret of its usefulness. The Joukowski map (z + 1/z)/2, by contrast, has derivative (1 - 1/z^2)/2 vanishing at z = plus or minus 1, and those two critical points are exactly what fold its circles into the sharp trailing edge of an airfoil. Knowing where conformality fails is knowing where each tool may and may not be pointed.