Trip on the Complex Plane

The Branch Cut

The complex plane may seem a peaceful place, with it's unit circle where numbers keep going round instead of flying to some infinity. But it is also a place full of surprises. If numbers go round, must there not be a point where they meet their own self?

Say you multiply the complex exponential e⁽⁰⁺ⁱ⁾, which is eⁱ, several times with itself. That is easy, as it is already expressed as an imaginary power of e, so you can double, triple, quadruple etc the exponent:

The same thing can be done with e^(0-i) and it's powers.

Negative powers mix with positive powers. They have no separate domain.

The overlap will even extend more when multiplication is continued beyond one cycle, with powers eⁱ⁷, eⁱ⁸ etcetera, and their negative counterparts. Before you know it, you have your whole complex plane messed up with all sorts of frequencies mingled beyond distinction.

Complex vectors (x+iy) all have their unique location on the complex plane, but when expressed as complex exponentials, that is no longer the case. An exponential e^iß is on the same location as e^i(ß+2pi) and infinitely many others. They are periodical modulo 2pi.

This overlap is an inescapable fact. In some cases it is advantageous. But it also relates to some tough problems in digital signal processing, such as aliasing and imaging.

To track what is exactly happening, it can be fruitful to regard the complex plane not as just one plane, but a whole stack of complex plane sheets.

Such sheets they call branches. Everytime you go more than a full cycle, you trespass a branch cut somewhere, and land on a next branch. Or a previous, depending on your direction.

Where is that branch cut, what does it look like, how do you pass from one branch to another?

The branch cut will be right through the negative x-axis, from the origin to minus infinity. At least for dsp that is the best place to do it.

Instead of endlessly going round on the same plane, there is now something like a smooth spiral staircase of planes where you go up or down. Depending on the type of operation, trespassing the branch cut may be equivalent to entering the alias range.

I have tried to model such a spiral complex plane from sheets of paper. But it is impossible. The sheets will be stuck together tightly at the point of the origin, so the whole thing cannot be stretched to a 3D object.

Anyway, the stack-of-sheets-interpretation in itself does not alter the numbers, or the overlap issue. Rather, it is an intro to complex logarithms. And these do transform the complex plane in a spectacular way. That is for a next episode.