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Newton Basin Project

egoughnour
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Previous: Colored By Iterations

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 $g(z;k) \equiv f(z)+u(|f(z)|-k-1) k Sin(z)$. 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 $z$, $sinh(\Im z)$. Proof of which.

Is 7 in the basin of attraction of any roots of the function f(z)?
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