# Complex Exp

Complex · w = f(z, t)

`w = exp(z · exp(i · t · 0.3))`

[Open in the app](https://www.wavelace.com/app#p=54) · [This page](https://www.wavelace.com/presets/complex-exp)

### The exponential

Read `exp(z · exp(i · t · 0.3))` from the inside out. `exp(i · t · 0.3)` has modulus 1 whatever the clock reads, so the inner product only turns `z` about the origin, by `0.3` radians per second at `Speed` 1. Set it aside for a moment and the function is `exp(z)`, whose two halves come apart cleanly: `|w| = e^x` depends on the real part alone and `arg w = y` on the imaginary part alone.

### Why it is a plaid

The hue repeats every `2π ≈ 6.28` along the imaginary direction, in bands running parallel to the real axis. The plot is 8 wide at the opening `Span of z`, so a little over one full sweep of the colour wheel fits. The brightness bands fall where `x/log(2)` is a whole number, straight lines `log(2) ≈ 0.693` apart crossing the hue bands at a right angle.

The relief is all on one side. At `x = 4`, `|w| ≈ 54.6` and the height is `0.6 · log(55.6) ≈ 2.41`. At `x = −4`, `|w| ≈ 0.018` and the sheet lies flat on the ground. A cliff on the right, a plain on the left, and no zero anywhere: `e^x` is never `0`, so the exponential never touches the ground and never has a pole either.

Restore the rotation and the whole plaid turns rigidly about the origin, once every `2π/0.3 ≈ 20.9` seconds, dragging the cliff around with it.

### Try

- `Top`: the plaid seen flat on, with the two families of bands square to each other.
- Raise `Iterations` to 20 so the formula is fed its own output over and over. Every point whose orbit escapes drops to the outside tone, shaded by how soon it left.
- Halve the turn rate with `exp(z · exp(i · t · 0.15))`: a full revolution now takes `2π/0.15 ≈ 41.9` seconds.
- Compress the plaid with `exp(2z · exp(i · t · 0.3))`: the hue bands and the brightness lines both halve their spacing, and the cliff doubles in steepness.
- `exp(1/z)`: the same striping wound around an essential singularity at the origin, where every value but `0` is taken over and over in any disc you care to draw.

### Read more

- [Exponential function](https://en.wikipedia.org/wiki/Exponential_function)
- [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula)
- [Essential singularity](https://en.wikipedia.org/wiki/Essential_singularity)
- [Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring)