A map of the plane can stretch, bend and fold. Some maps, though, are remarkably gentle: at a
point they may rotate and rescale a tiny neighbourhood, but they leave every angle
intact. Such a map is called conformal, and the surprise of this stage is that
the holomorphic
functions you already know are exactly the conformal ones.
Precisely: a map f is conformal at
z_0 if, for any two smooth curves crossing at
z_0, their images under f cross at
f(z_0) at the same angle — same size and same
sense (orientation). It preserves angles; it generally does not preserve lengths.
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If f is holomorphic at z_0 and
f'(z_0) \ne 0, then f is
conformal at z_0: it preserves the angle
between any two curves through z_0, in size and in sense.
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Where f'(z_0) = 0 the map is not conformal —
angles are multiplied. At such a point z^2 doubles
every angle, z^3 triples it, and so on.
Why a nonzero derivative preserves angles
The whole proof is the local picture of the
complex
derivative together with the fact that
complex
multiplication is a rotate-and-scale.
Step 1 — linearise f near
z_0. Differentiability means that for
z close to z_0,
f(z) \approx f(z_0) + f'(z_0)\,(z - z_0).
So, up to the negligible higher-order part, the displacement
z - z_0 is sent to the displacement
f'(z_0)\,(z - z_0). Everything that happens to a tiny arrow leaving
z_0 is just multiplication by the one fixed complex number
f'(z_0).
Step 2 — read that multiplier in polar form. Write
f'(z_0) = |f'(z_0)|\,e^{i\varphi}, \qquad \varphi = \arg f'(z_0).
Multiplying any arrow by this number rotates it by the fixed angle
\varphi and scales its length by the fixed factor
|f'(z_0)|. This needs f'(z_0) \ne 0: if
the multiplier were 0 there would be no direction to speak of.
Step 3 — apply it to two directions at once. Let two curves leave
z_0 in directions making angles
\alpha and \beta. After the map their
tangent directions are
\alpha + \varphi \qquad \text{and} \qquad \beta + \varphi.
Step 4 — subtract. The same \varphi is added to
both, so it cancels in the difference:
(\beta + \varphi) - (\alpha + \varphi) = \beta - \alpha.
The angle between the two curves is exactly what it was — and because both turned by the same
\varphi in the same direction, the sense of the angle is
preserved too. That is conformality.
Where it fails: f'(z_0) = 0
At a point with f'(z_0) = 0 the linear term vanishes and the next
term takes over. For f(z) = z^2 at the origin,
f'(0) = 0 and
f(z) = z^2 = \big(r e^{i\theta}\big)^2 = r^2 e^{i\,2\theta}.
A ray leaving the origin at angle \theta is sent to a ray at angle
2\theta, so the angle between two rays is doubled.
Two perpendicular rays come out parallel. The map is angle-preserving everywhere
except at the one point where its derivative dies.
It is worth savouring how little a conformal map is required to do. It may stretch a region
enormously near one point and shrink it near another — the scale factor
|f'(z)| roams freely. The single quantity it guards is the angle
at which curves cross. That is why a fine square grid, pushed through a holomorphic map,
comes out as a warped mesh whose every intersection is still a perfect right angle. This
local rigidity is the engine behind everything in this stage: Möbius maps that send circles
to circles, and the trick of solving a physics problem on an awkward shape by reshaping it
conformally into an easy one.