the automorphisms of the upper half-plane
The upper half-plane is the set of complex numbers z = x + i y with positive imaginary part (y > 0) — everything strictly above the real axis. It is a twin of the unit disk: the two regions look different but are conformally identical, and so they share the same symmetries in disguise. The automorphisms of the upper half-plane are the holomorphic, reversible self-maps of this region, and they have a beautifully clean description in terms of real numbers.
Every conformal automorphism of the upper half-plane has the form f(z) = (a z + b)/(c z + d), where a, b, c, d are REAL numbers with a d - b c > 0 (positive determinant). These are exactly the Mobius maps with real coefficients and positive determinant. The reality of the coefficients keeps the real axis mapped to the real axis (with a point sliding off to infinity), and the positive determinant keeps the upper half above the line rather than flipping it below. Scaling all four coefficients together does not change the map, so the natural group is PSL(2,R): two-by-two real matrices of determinant 1, modulo plus-or-minus the identity. The translation z -> z + b (real b), the dilation z -> a^2 z (a real), and the inversion z -> -1/z generate the whole group.
Why bother with the half-plane when we have the disk? Because for many computations the half-plane is friendlier — its symmetry group is the concrete real matrix group PSL(2,R), which is the gateway to the modular group SL(2,Z) and hence to modular forms, elliptic curves, and number theory. The Cayley transform z -> (z - i)/(z + i) is the explicit dictionary translating disk automorphisms into half-plane ones and back. An honest caution: the coefficients must be real and the determinant positive. Real coefficients with NEGATIVE determinant map the upper half-plane to the LOWER half-plane (an anticonformal-like flip in effect); complex coefficients break the real axis entirely.
The map f(z) = -1/z has coefficients a=0, b=-1, c=1, d=0 with determinant ad - bc = 1 > 0, so it is an automorphism of the upper half-plane: it sends i to i (a fixed point), and sweeps the imaginary axis to itself reversed. Combined with translations z -> z + 1, these two generate the modular group SL(2,Z) acting on the half-plane — the source of modular forms.
z -> -1/z and z -> z + 1 generate the modular group on the half-plane.
It is a common error to allow complex coefficients here. For the half-plane the coefficients must be REAL with positive determinant. (For the disk, by contrast, the natural matrix group is PSU(1,1), with complex entries of a special conjugate-symmetric shape.) Same abstract group, different concrete realizations.