# Newton Basin Project

egoughnour
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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 $g\left(z;k\right)\equiv f\left(z\right)+u\left(|f\left(z\right)|-k-1\right)kSin\left(z\right)$. This will yield the same roots as $f\left(z\right)={x}^{3}-x$ on the real line. This obtains from Rouché's theorem as long as $|f|, $sin\left(z\right)$ being of bounded modulus there. Elsewhere the value can be seen as depending on the hyperbolic sine of the imaginary part of $z$, $sinh\left(\mathrm{\Im }z\right)$. Proof of which.