ISEGORIA / MATH ENCYCLOPEDIA
Complex analysis: maps, residues, and harmonic fields
Conformal maps, residues, and harmonic functions.
Before you begin: Calculus and complex numbers
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Conformal maps
Transform a grid with w = zᵏ and follow two short curves through a point z₀ that you drag. Away from z₀ = 0 both curves are rotated by arg f′(z₀) and stretched by |f′(z₀)|, so the angle between them is unchanged. At the critical point z₀ = 0 the angle is multiplied by k.
Worked example. The derivative multiplies by a local scale and rotation.
Watch out. The map is not angle-preserving at z=0, where the derivative vanishes.
What does the derivative control?
Its magnitude scales lengths and its argument rotates directions.
2. Residues around a pole
The function f(z) = 1/(z − a) + 2/(z − b) has residue 1 at a and 2 at b. Drag the centre of the circular contour, change its radius and move the pole a. The right panel traces the integral accumulated along the way: it returns to 0 when no pole is enclosed and ends at 2πi times the enclosed residues otherwise, jumping when a pole crosses the contour.
Worked example. A contour enclosing one simple pole contributes 2πi times its residue.
Watch out. The contour must avoid poles and be positively oriented for this sign convention.
What changes when the pole crosses the contour?
The enclosed residue sum changes discontinuously.
3. Harmonic conjugates
Level curves of u = Re f (blue) and v = Im f (orange) for a holomorphic f, rotated by the phase φ. Drag the probe: the two gradients are always perpendicular and equally long, as the Cauchy–Riemann equations demand, except at critical points where f′ = 0.
Worked example. The real and imaginary parts of z² are x²−y² and 2xy.
Watch out. Orthogonality refers to regular level curves, away from critical points.
Why are the families orthogonal?
The Cauchy–Riemann equations rotate one gradient into the other.