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Complex numbers and conformal mapping

What if I told you that every analytic function in complex numbers is a conformal mapping?

This wouldn't be very helpful since you either know this already or have no idea what am I talking about. If you want, I can explain it in a very simple but probably not entirely correct manner. It'll only take five minutes.

So complex numbers are numbers with imaginary parts. Numbers like this.

a + bi, i² = -1

Let's pretend they are something else. Let's pretend they are vectors. The real part then shall go as the usual x-axis, and the imaginary part — as the y-axis.

Summing them up works just like with vectors: real parts sum up separately from the imaginary parts, just like x-parts — separately from y-parts in vectors.

(a + bi) + (c + di) = (a + c) + (b + d)i

Here is an interactive plot. Feel free to move red and blue arrows around. The barber-shop arrow is the sum, it's made unmoveable by itself.

Complex multiplication is somewhat different from what we usually do with vectors. The formula doesn't look too inspiring.

(a + bi) * (c + di) = (ac - bd) + (bc + ad)i

However, geometrically, multiplying a complex number to a normalized complex number is a rotation.

Ok, but what if the complex multiplier isn't normalized?

We can normalize it by dividing by its length and then multiply the result by this length. Multiplying a complex number by a scalar is just like scaling a vector.

c*(a + bi) = ca + cbi

So, additions and multiplications are all translations, rotations, and scales. There is no shear. These three transformations preserve the angles locally. And preserving angles locally is what constitutes a conformal transformation in geometry. In the analysis, transformations are usually called maps. It's just a convention. “Conformal map” and “conformal transformation” is the same thing.

And now for analytic functions. Their definition might not be too helpful for our case, but they have a property that is.

The property is: a function is analytic if and only if its Taylor series about x₀ converges to the function in some neighborhood for every x₀ in its domain.

The Taylor series is a polynomial series, and polynomes consist only of additions, multiplications, and scales.

So if a function acts as a polynomial locally, it can only move, rotate, and scale complex-numbers made vectors, preserving the angles locally, therefore, forming a conformal transformation or, which is the same, a conformal map.

Here is the living example. It transforms a grid from [-1, 1]×[-1, 1] to ℂ. Please note that all the grid sections always intersect by the right angle no matter how twisted the transformation is.

Feel free to propose your own transformation. The formula box from below accepts + − * / ( ) and ^ as “power”.

There is also a t variable that oscilates in [-1, 1].

I hope you now have a good understanding of conformal mapping and the geometry of complex numbers in general. Did it take the full 5 minutes?