iterated derivatives of arctan

Stream: Is there code for X?

[Kim Morrison]

In case anyone is looking for a project, I'd love to have the iterated derivatives of arctan available.

Something like:

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$\frac{d^n}{dx^n}(\arctan(x)) = (-1)^{n-1}(n-1)! \sin^n(\theta) \sin(n\theta)$

where

$\theta = \arctan\left(\frac{1}{x}\right)$

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$\frac{d}{dx}(\arctan(x)) = \frac{1}{1+x^2} = \frac{1}{2i}\left(\frac{1}{x-i} - \frac{1}{x+i}\right)$
$\frac{d^n}{dx^n}(\arctan(x)) = \frac{(-1)^{n-1}(n-1)!}{2i}\left(\frac{1}{(x-i)^n} - \frac{1}{(x+i)^n}\right)$

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Theorems describing the iterated derivatives evaluated at 0, as consequences of one of these (probably the second).

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A formula for taylorWithinEval arctan n (Set.Icc a b) 0 y (here 0 and y are in the interior of Set.Icc a b) in terms of ∑ i ∈ Finset.range (n + 1), (iteratedDeriv_arctan_zero i) * x ^ i / i !