Conformal maps and local geometry

Try

Near z=1, the map z² looks to first order like multiplication by two, so tiny shapes are approximately doubled without angle distortion. Near z=i, it looks like multiplication by 2i: a doubling and quarter-turn. At zero, the derivative vanishes and this first-order description collapses.

Conformal here means locally preserving oriented angles through a nonzero complex derivative. The distinction between local behavior and global one-to-one mapping is essential. The exponential has nonzero derivative everywhere but repeats values after vertical shifts of 2πi.

Predict

Predict what z² does to a right-angle sector based at zero. It doubles the argument, turning a quadrant into a half-plane. At the vertex it does not preserve the angle, even though away from zero it is conformal.

Predict whether openness requires a nonzero derivative. It does not: z² still maps a disk centered at zero onto a disk. Local invertibility is the stronger property that fails at the critical point.

Experiment

Compare local derivative data.

Map and point	Derivative	Length scale	Rotation
z² at 1	2	2	0
z² at i	2i	2	π/2
z² at 0	0	Degenerate first order	Not defined

These are infinitesimal quantities. A finite region may bend substantially, so a derivative scale at one point is not a global distance formula.

Understand

Let two regular differentiable curves pass through a, with nonzero tangent vectors v and w. Under a holomorphic f, their image tangents are f′(a)v and f′(a)w. If f′(a)≠0, their ratio remains v/w, so the oriented argument difference is unchanged. Each speed is multiplied by |f′(a)|. In real coordinates the derivative has determinant |f′(a)|²>0, so orientation is preserved.

The real inverse function theorem gives a local inverse when this determinant is nonzero. Its derivative is multiplication by 1/f′, so it is holomorphic. A biholomorphic map is a holomorphic bijection with holomorphic inverse; it supplies a global conformal equivalence between domains.

For a concrete global example, z² maps the open sector 0<arg z<π/2 bijectively to the upper half-plane. Radius squares and argument doubles, and each target has a unique half-angle in the sector. Zero is excluded, so the derivative never vanishes there.

For |a|<1, the disk map φ_a(z)=(z−a)/(1−āz) satisfies |1−āz|²−|z−a|²=(1−|a|²)(1−∣z∣²). The right side is positive inside the disk, proving |φ_a(z)|<1. Its denominator is nonzero there. The inverse (w+a)/(1+āw) obeys the same kind of bound, proving onto. Multiplying by a unit complex number adds a disk rotation.

Apply

Restrict exp z to the strip −π<Im z<π. Equal outputs differ by 2πik, but two points in that open strip have imaginary difference strictly between −2π and 2π. Hence the only possible integer is zero, proving injectivity. Its image is the principal slit plane.

Repair “A conformal map preserves all lengths.” It preserves angles locally and scales lengths infinitesimally by a factor that can depend on the point. Even a simple nonunit dilation changes every distance.

Continue

Harmonic functions connect these maps with real-valued potential problems. Optional depth: investigate how a conformal map changes a boundary corner using a power map. The Riemann mapping theorem gives a much broader existence result for simply connected proper plane domains, but its proof is beyond the explicit mapping constructions developed here.