Question 1

What is domain colouring?

Accepted Answer

Domain colouring (Frank Farris / Hans Wegert) represents complex functions f: ℂ→ℂ as images. Each point z in the complex plane is coloured by a property of f(z): hue encodes the argument (angle) of f(z) — cycling through the rainbow as arg f(z) goes from −π to π, while brightness encodes the magnitude |f(z)|. Contour rings mark powers of two in |f(z)|, so zeros appear as bright centres where all hues converge and poles appear as dark pits where colours swirl in the reverse orientation.

Question 2

How can I identify zeros and poles visually?

Accepted Answer

A zero of order n appears as a point where all n rainbow cycles converge — the hues wind around n times as you circle the zero. A simple zero (n=1) shows one full rainbow cycle around it. A simple pole shows one cycle in the reverse (clockwise) direction. Essential singularities (like at ∞ for sin and exp) show infinitely many colour cycles compressing together. Branch cuts appear as sharp discontinuities in hue.

Question 3

What is conformal mapping?

Accepted Answer

A holomorphic function f with f'(z)≠0 is conformal: it preserves local angles between curves. In domain colouring this manifests as 90° grid squares mapping to 90° (locally square) image regions. Möbius transformations (az+b)/(cz+d) are globally conformal on the Riemann sphere ℂ̂; they map circles and lines to circles and lines. The visualizer lets you compare how a function distorts the plane: look for where the image looks locally like scaled rotations versus where distortion is large.

Question 4

Why does exp(z) look like vertical stripes?

Accepted Answer

Because exp(z) = exp(x+iy) = eˣ · (cos y + i·sin y): the argument cycles with period 2π in the imaginary direction (vertical) while the magnitude grows exponentially in the real direction (horizontal). So horizontally you see the magnitude change (brightness varies), and vertically you see one complete rainbow per 2π strip. The function has no zeros but an essential singularity at ∞ in the left half-plane as exp(z)→0.

🧪 Complex Function Visualizer

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