the inversion map 1/z
The inversion map is f(z) = 1/z, the reciprocal. In polar form, if z = r e^(i theta) then 1/z = (1/r) e^(-i theta). So it does two things: it inverts the distance from the origin (a point at distance r goes to distance 1/r), and it reflects across the real axis (the angle flips sign). Points far out come in close, points near the origin fly far out, and the unit circle |z| = 1 maps to itself.
Its signature property is that it sends lines and circles to lines and circles — provided you agree to treat a line as a 'circle through the point at infinity'. A circle not passing through the origin maps to another circle; a circle through the origin maps to a straight line; a line through the origin maps to a line; a line missing the origin maps to a circle through the origin. To work an image you substitute z = 1/w into the equation of the curve and simplify. The origin itself has no image (you cannot compute 1/0), which is exactly why this map is best understood on the Riemann sphere, where 0 maps to infinity and infinity maps to 0.
Inversion is conformal away from the origin and is one of the basic building blocks — together with translation and rotation-scaling — out of which every Mobius transformation is assembled. Its circle-preserving magic is what makes Mobius maps the natural language for moving disks, half-planes, and circular arcs around, and it is the engine behind many a clever change of variables in physics and integral evaluation.
Under f(z) = 1/z, the vertical line Re(z) = 1/2 maps to the circle |w - 1| = 1 (a circle through the origin), and the exterior of the unit circle |z| > 1 maps to the punctured interior 0 < |w| < 1.
Inversion swaps inside and outside of the unit circle and turns many lines into circles.
1/z is undefined at z = 0, so as a plane-to-plane map its domain excludes the origin. Adjoining the point at infinity (working on the Riemann sphere) repairs this: there 1/z swaps 0 and infinity and becomes a clean one-to-one map of the whole sphere.