Newton Basin Project

Here is a slightly more interesting output

Note that $$ u$$ here is the Heaviside step function but it's applied to the (shifted) modulus of a function rather than a variable. That is, we are interested in roots of $$ g(z;k)\equiv f(z)+u(|f(z)|-k-1)kSin(z)$$. This will yield the same roots as $$ f(z)=x^3-x$$ on the real line. This obtains from Rouché's theorem as long as $$ |f|<k$$, $$ sin(z)$$ being of bounded modulus there. Elsewhere the value can be seen as depending on the hyperbolic sine of the imaginary part of $$ z$$, $$ sinh(\mathrm{\Im }z)$$. Proof of which.