Desmos now supports built-in complex number functions, making it easier to visualize how functions transform the complex plane.

The example above shows the function:

$$ f(z) = z^{t} $$

applied to a grid in the complex plane. You can see how the structure of the plane is warped by the function — lines become curves, but angles are preserved, which is a key property of conformal mappings.

Functions like $z^n$ (where $n$ is a real or complex exponent) exhibit interesting behavior due to branch points and multi-valuedness. For instance, fractional powers like $z^{4/3}$ introduce multiple “sheets” — each corresponding to a different branch of the complex logarithm.

The Desmos calculator contains many custom-defined complex functions in this graph — but thanks to recent updates, most of these can now be replaced with Desmos’ native complex operations.