# Complex Sine

Complex · w = f(z, t)

`w = sin(z)`

[Open in the app](https://www.wavelace.com/app#p=55) · [This page](https://www.wavelace.com/presets/complex-sine)

### The zeros on the real axis

Split `z` into `x + iy` and `sin(z)` comes apart: `sin(z) = sin(x) · cosh(y) + i · cos(x) · sinh(y)`, so `|w|² = sin(x)² + sinh(y)²`. Along the real axis, where `y = 0`, that is the familiar sine, never larger than 1, and it vanishes at every multiple of `π`.

Three of those zeros are on the plot, at `0` and `±π ≈ ±3.14`. The next pair, at `±2π ≈ ±6.28`, is outside the opening `Span of z` of 4. Each is a simple zero, so the sheet dips to the ground there and the colour wheel makes exactly one anticlockwise turn around it. Neighbouring zeros are half a turn out of step with one another, because near `z = kπ` the function is close to `±(z − kπ)` with the sign alternating.

### A valley and two cliffs

Away from the real axis `sinh(y)` takes over and the sheet climbs hard. At `y = ±4`, `|w| ≈ 27.3` and the height is `0.6 · log(28.3) ≈ 2.01`, near enough `e^|y|/2`. The bound of 1 that the real sine obeys holds on one line and nowhere else.

That is forced: the sine is defined and differentiable everywhere, and by Liouville's theorem the only such functions that stay bounded are the constants. The whole figure repeats every `2π` along the real direction, a valley of evenly spaced zeros between two rising walls.

### Try

- `Top`: the valley flat on, with the three wheels strung along the real axis.
- `cos(z)`: the same landscape shifted by `π/2 ≈ 1.57`, its zeros at `±1.57` instead.
- `sin(z)/z`: the zero at the origin cancels against the one below it and the sheet lifts to `|w| = 1` there, leaving the other two dimples in place.
- `tan(z)`: the zeros stay put and simple poles appear between them, at `±π/2 ≈ ±1.57`, with the wheel running backwards around each.
- Raise `Iterations` to 20 so the sine is fed its own output over and over. The walls escape at once and take the outside tone, while the valley floor is drawn back toward the zero at the origin and stands on a flat plateau.

### Read more

- [Trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions)
- [Entire function](https://en.wikipedia.org/wiki/Entire_function)
- [Liouville's theorem (complex analysis)](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis))
- [Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring)