Mathematical Analysis · Complex Analysis
Multiplying by a complex number is a rotational homothety (material «Complex Numbers: Three Constructions of One Field»); the derivative is the same limit as for real functions (material «Regular Functions»). Here — what follows from this geometrically: regularity turns out to be a similarity "in the infinitesimal," and so preserves angles between curves, and the inversion $w=1/z$ lets us see and prove this in its purest form.
Let $f$ be differentiable at the point $z_0$, and let $a=f'(z_0)$. By the definition of the derivative,
$$f(z)=f(z_0)+f'(z_0)(z-z_0)+o(z-z_0)\quad\text{as }z\to z_0,$$
that is, near $z_0$ the function $f$ is approximated by its linear part
$$f(z)\approx f(z_0)+a(z-z_0)=az+\big(f(z_0)-az_0\big).$$
This is exactly the similarity $w=az+b$ from the material «Complex Numbers: Three Constructions of One Field» with $b=f(z_0)-az_0$ — a rotational homothety with coefficient $|a|$ and rotation angle $\arg a$. The only difference from that material: there, $a$ was a fixed number for the whole transformation of the plane; here $a=f'(z_0)$ varies from point to point — this is a similarity "in the infinitesimal," its own at every point of the domain of regularity.
Let a smooth curve $\gamma(t)$ pass through the point $z_0$, with $\gamma(0)=z_0$ and tangent vector $\gamma'(0)=\tau\ne0$. The direction of the curve at $z_0$ is $\arg\tau$; the angle between two curves intersecting at $z_0$ with tangents $\tau_1,\tau_2$ is the difference $\arg\tau_2-\arg\tau_1$.
Let's see what happens to this angle under a regular function $f$ with $f'(z_0)=a\ne0$. The image of the curve is $f(\gamma(t))$, and its tangent at $t=0$ is:
$$(f\circ\gamma)'(0)=\lim_{t\to0}\frac{f(\gamma(t))-f(\gamma(0))}{t}=\lim_{t\to0}\frac{f(\gamma(t))-f(z_0)}{\gamma(t)-z_0}\cdot\frac{\gamma(t)-z_0}{t}=a\tau.$$
The first factor tends to $f'(z_0)=a$, because $\gamma(t)\to z_0$ as $t\to0$, and the complex derivative is a limit along any direction of approach (material «Regular Functions», §1), not only along a straight line; the second factor tends to $\gamma'(0)=\tau$ by definition.
So the tangent to the image of the curve is $a\tau$: the original tangent $\tau$, multiplied by the same number $a$, for any curve through $z_0$. For two curves with tangents $\tau_1,\tau_2$, the angle between the images is
$$\arg(a\tau_2)-\arg(a\tau_1)=\big(\arg a+\arg\tau_2\big)-\big(\arg a+\arg\tau_1\big)=\arg\tau_2-\arg\tau_1$$
— the same angle as between the original curves, with the same sign (orientation is preserved too). This is exactly conformality.
If $f$ is regular at the point $z_0$ and $f'(z_0)\ne0$, then $f$ is conformal at $z_0$: it preserves angles between curves, including the sign of the angle.
The condition $f'(z_0)\ne0$ matters: where the derivative vanishes, the linear part of the approximation from §1 disappears, and angles are, in general, not preserved. A simple example: $f(z)=z^2$ at the point $z_0=0$. Writing $z=re^{i\theta}$, we get $f(z)=r^2e^{2i\theta}$ — the angle between any two rays from zero is doubled by $f$, not preserved.
The simplest example of a regular function that isn't just a similarity is inversion, $w=1/z$. It is regular and conformal everywhere except $z=0$: $w'(z)=-1/z^2\ne0$ for $z\ne0$. It has a famous property — it sends circles and lines back to circles and lines.
Let the circle $|z-c|=r$ not pass through zero ($|c|\ne r$). Then its image under $w=1/z$ is a circle with center $c'=\bar c/(|c|^2-r^2)$ and radius $r'=r/\big||c|^2-r^2\big|$.
The equation of the circle $|z-c|=r$ can be written as $(z-c)(\bar z-\bar c)=r^2$, that is, $z\bar z-c\bar z-\bar cz+|c|^2-r^2=0$. Substitute $z=1/w$, $\bar z=1/\bar w$ (this is exactly the inverse transformation of $w=1/z$) and multiply by $w\bar w\ne0$:
$$1-cw-\bar c\bar w+(|c|^2-r^2)\,w\bar w=0.$$
Divide by $|c|^2-r^2\ne0$:
$$w\bar w-\frac c{|c|^2-r^2}w-\frac{\bar c}{|c|^2-r^2}\bar w+\frac1{|c|^2-r^2}=0.$$
The equation of the circle $|w-c'|=r'$ in the same notation is $w\bar w-\bar{c'}w-c'\bar w+|c'|^2-r'^2=0$. Comparing the coefficients of $w$ and $\bar w$ gives $\bar{c'}=c/(|c|^2-r^2)$, which is where the formula for $c'$ comes from; from the constant term, $|c'|^2-r'^2=1/(|c|^2-r^2)$, which after substituting $|c'|^2$ gives the formula for $r'$.
(If the circle does pass through zero, $|c|=r$, the denominator vanishes, and the image is no longer a circle but a line; this limiting case isn't shown in the interactive widget below — the circle's center is deliberately kept from approaching zero closer than a fixed bound, so that $1/z$ never has to be computed arbitrarily close to the singular point.)
The function $\displaystyle w=\frac{az+b}{cz+d}$, $a,b,c,d\in\C$, $ad-bc\ne0$, is called a fractional-linear (or Möbius) transformation. The condition $ad-bc\ne0$ is exactly the condition that keeps $w$ from degenerating into a constant function.
When $c=0$, the transformation is just $w=(a/d)z+(b/d)$, a similarity; circles and lines map to circles and lines trivially (material «Complex Numbers: Three Constructions of One Field»).
When $c\ne0$, divide the numerator by the denominator:
$$\frac{az+b}{cz+d}=\frac ac+\frac{bc-ad}{c^2}\cdot\frac1{z+d/c}$$
(check by expanding: $\frac ac(cz+d)+\frac{bc-ad}c=az+\frac{ad}c+\frac{bc-ad}c=az+b$). The right-hand side is: a translation by $d/c$, then inversion, then a rotational homothety with coefficient $(bc-ad)/c^2$ (nonzero exactly when $ad-bc\ne0$), then a translation by $a/c$.
Translation and similarity send circles and lines to circles and lines trivially (material 1); inversion does so by the theorem in §3. A composition of transformations with this property again has this property, so any fractional-linear transformation sends circles and lines to circles and lines too — with no separate proof needed for the general case.
As a composition of regular functions (translation, inversion, and similarity are all regular and conformal wherever defined), a fractional-linear transformation is regular and conformal on its entire domain — that is, everywhere except the point $z=-d/c$ (for $c\ne0$), where the denominator vanishes.
Let's check this not on inversion, but on a genuine fractional-linear transformation with nonzero $a$. Take the classic example $w=\dfrac{z+1}{z-1}$ (the unit circle $|z|=1$ maps to the imaginary axis — this can be checked by direct substitution $z=e^{i\theta}$) and apply the same trick as in §3: sample the circle at 120 points and plot the image.
The three materials in this series traced a path from algebra to geometry, by way of analysis. First — three equivalent constructions of the field ℂ, and the discovery that multiplying by a number is a rotational homothety (material «Complex Numbers: Three Constructions of One Field»). Then — what it means to be a differentiable function of a complex variable, and why that condition turns out to be surprisingly rigid: regularity implies representability as a Taylor series, uniqueness of continuation, an integral theorem, and Cauchy's formula (material «Regular Functions»).
And finally, here — what regularity means geometrically: at every point where the derivative is nonzero, the function behaves like a similarity "in the infinitesimal," and so preserves angles between curves. Inversion and its descendants, fractional-linear transformations, show this property in its purest form — in an example that can be carried explicitly from definition to finished formula.