complex functions as mappings
An ordinary function y=f(x) takes one number on the number line and returns one number, so you can draw its graph. A complex function w=f(z), however, takes one point on a plane and returns one point on a plane. Input and output are both 2D, so an ordinary graph cannot show it. Instead, think of the function as kneading the whole input plane by hand into a different shape. Straight grid lines bend, sizes change, inside and outside swap. Yet a remarkable rule hides in this deformation. However much it bends, a differentiable complex function keeps angles unchanged on a small enough scale. This is called a conformal map, and it makes a powerful tool: a problem with an awkward shape can be turned into one with an easy shape and solved there. In this lesson you reshape the plane yourself and see how a function kneads it.
First, see what it means for a function to deform the plane. Lay a straight square grid over the input plane and choose a function: where that grid ends up is drawn as gold curves. Choose z² and the grid bends smoothly into a curved grid; choose 1z and the region near the origin turns inside out; choose ez and the grid fans out. The buttons below switch among the three. The key point is that straight lines become curves. The function has bent the plane. Not "put in x and get out y" as with an ordinary function, but "reshape the whole plane at once": that is the way to look at a complex function.
Look closely at the most basic deformation, z². Drag the blue point z and the gold z² moves with it. The rule is one you already know from the last two lessons. In polar form, if z = r∠θ then z² = r²∠2θ. The angle doubles and the size is squared. The blue arc and the gold arc next to the origin show the two angles, and the two lines below give their values. Put θ at 30 degrees and z² is at 60 degrees; put θ at 90 degrees and z² is at 180 degrees. This bends the grid because every point's angle opens up twofold, so straight lines fan out and curve. In the last lesson, multiplying by eiθ twice added the angle twice. z² is z multiplied by itself, so its angle being θ+θ=2θ is only natural. A map is, in the end, multiplication applied to the whole plane at once.
Now for the true jewel of mapping: conformality. We said z² bends the grid, yet look at where the horizontal and vertical lines through z0 end up: both have curved, but they still meet at exactly 90 degrees. The two faint straight lines are the input; the thick green and orange curves are where they went, and the crossing angles of input and output appear side by side below. Drag z0 anywhere: two lines that crossed at a right angle in the input still cross at a right angle after the deformation. That is conformality. However much a function bends and stretches the plane, zoom in close enough to one point and only rotation and scaling take place. As "Multiplying Complex Numbers Is Rotating" showed, rotation and scaling do not change angles, so angles are preserved. It fails at exactly one place, the origin, because the derivative of z², 2z, is 0 there. Bring the point near the origin and that warning appears. Apart from that exception, a differentiable complex function keeps angles everywhere.
Now a different kind of deformation: 1z. Drag the blue point z and the gold 1z moves with it, in a rather peculiar way. 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. Points inside the dashed unit circle (radius 1) go outside, and points outside come inside. A point hugging the origin flies far out toward infinity, while a very distant point gathers near the origin. The unit circle acts as a mirror that swaps inside and outside. The change of angle to −θ is a mirror reflection across the real axis. This inside-out map is conformal too, so right angles stay right angles. In electric-field or flow problems where the far-away region is awkward to handle, 1z turns it inside out into a finite problem near the origin.
Finally, see why engineers prize this so much. Finding how water flows around a cylinder directly is hard. A uniform flow with no obstacle, just parallel straight streamlines, is very easy. The magic of conformal mapping is that it connects the two. Switch between the "Uniform flow" and "Flow around cylinder" buttons and you see a single map, w=z+1z, carry the straight streamlines of the uniform flow exactly onto the gold streamlines that wrap around the blue cylinder. So in practice you solve the easy case and then simply carry the answer over to the hard shape with the map. Because conformality preserves angles, physical properties such as the angle at which a streamline meets a boundary survive the trip, and the carried-over answer really is correct. Airflow around an aircraft wing, the electric field around an electrode, and water flowing under a dam are all solved this way. That is why complex numbers are not mere abstraction but a working tool of engineering.