$$ \begin{align*} x &= \mathrm{erf}\ \mathrm{atanh}\ \mathrm{cos}\ t \ y &= \mathrm{erf}\ \mathrm{atanh}\ \mathrm{sin}\ t \end{align*} $$

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proof notes

let us say that f(t) = erf(cot(t)) h(t) = erf t h'(t) = 2 e^(-z^2) / √π g(t) = cot t g'(t) = csc² t

the chain rule states that f'(x) = h'(g(t))g'(t) so f'(t) = (2 / √π) e^(-cot² t) csc² t

the product rule states that (uv)' = u'v + uv' so u' = (e^(-cot² t))' = -2 cot(t) csc²(t) e^(-cot² t)

u' = -2 cot(t) csc²(t) u

f⁽ⁿ⁾(t) ∝ u ∝ e^(-cot² t)

therefore f(t) is flat wherever e^(-cot² t) = 0

limit of e^(-cot² t) as t -> 0 = 0

limit of e^(-cot² t) as t -> nπ = 0

f(t) is asymptotically flat at nπ where n is an integer