A Complex Function Deforms the Plane into a Plane

First, let's see what it means for a function to deform the plane. I'll lay a straight graph-paper grid over the input plane. Pick a function, and you'll see where that grid gets carried to — its deformed image drawn in bold. Choose z² and the grid bends smoothly into a curved grid; choose 1z and the region near the origin gets flipped to the outside; choose e^z and the grid fans open. The key is that what was straight becomes curved. The function has bent the plane. Not "put in x, get out y" like an ordinary function, but "take the whole plane and deform it into a different shape" — that's the eye for seeing a complex function.

Of all the deformations, look closely at the most basic one, z². Drag the point z and z² moves with it. The rule is one your hands already know from the last two lessons. In polar form, if z = r∠θ, then z² = r²∠2θ. The angle doubles and the size squares. Put θ at 30° and z² is at 60°; put θ at 90° and z² is at 180°. Why does this bend the grid? Because every point's angle spreads to double, so a straight line fans open and curves. In lesson 2 we said multiplying e^iθ by itself adds the angle twice. z² is just z multiplied by itself, so of course the angle becomes θ+θ=2θ. A mapping is, in the end, multiplication applied to the whole plane at once.

Now for the real jewel of mappings: conformality. We said z² bends the grid. Yet look at the deformed images of a horizontal line and a vertical line through z0 — both have curved, and still they meet at exactly 90°. Drag z0 anywhere. Two lines that crossed at a right angle in the input still cross at a right angle after the deformation. This is conformality. No matter how much the function bends and stretches the plane, zoom in close enough near any one point and all that happens is rotation and scaling. Rotation and scaling don't change angles (you saw this in lesson 1). So angles are preserved. It breaks at exactly one place — the origin — because z²'s derivative 2z is zero there. That one exception aside, a differentiable complex function keeps angles everywhere.

Look at a different kind of deformation, 1z. Drag the point z and 1z moves with it, but the motion is peculiar. In polar form, 1z flips the size from r to 1r and changes the angle from θ to −θ. Flipping the size is the key. A point that was inside the unit circle (radius 1) goes outside, and one that was outside comes inside. A point hugging the origin flies off toward infinity, and a very distant point gathers near the origin. The unit circle acts as a mirror that swaps inside and outside. The angle becoming −θ is a mirror reflection across the real axis. This inside-out mapping is also conformal, so right angles are still preserved as right angles. When a field or flow problem makes "infinitely far away" awkward to handle, you flip it with 1z into a finite problem near the origin and solve that instead.

Finally, see why engineers treasure this so much. Working out how water flows around a cylinder directly is awkward. But a uniform flow with no obstacle at all — just parallel straight streamlines — is dead easy. The magic of conformal mapping is that it links the two. Toggle between the two pictures. A single mapping, w=z+1z, carries the easy uniform flow exactly onto the hard flow around a cylinder. So in practice you find the answer on the easy side, then just carry it over with the mapping to the hard shape, and you're done. Because conformality preserves angles, physical properties like the angle at which streamlines meet a boundary carry over intact, so the answer is genuinely correct. Airflow around an aircraft wing, the electric field around an electrode, water flow under a dam — all of it is solved this way. This is why complex numbers aren't just an abstraction but a working tool of engineering.