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Question Number 136370 by mathmax by abdo last updated on 21/Mar/21

letf (x) = x^{3} \arctan (\frac{π}{x})

1) calculate f^{(n)} (x)

2) calculate f^{(n)} (1)

3) developp f at integer serie

Answered by mathmax by abdo last updated on 22/Mar/21

1) f (x) = x^{3} \arctan (\frac{π}{x}) if x > 0 \Rightarrow f (x) = x^{3} (\frac{π}{2} - \arctan (\frac{x}{π}))

= \frac{π}{2} x^{3} - x^{3} \arctan (\frac{x}{π}) = u (x) - v (x) \Rightarrow f^{(n)} = u^{(n)} - v^{(n)}

u (x) = \frac{π}{2} x^{3} \Rightarrow u^{(1)} x) = \frac{3}{2} π x^{2}, u^{(2)} (x) = 3 π x and u^{(3)} (x) = 3 π

v (x) = x^{3} \arctan (\frac{x}{π}) \Rightarrow v^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(x^{3})}^{(k)} {(\arctan (\frac{x}{π}))}^{(n - k)}

= C_{n}^{0} x^{3} {(\arctan (\frac{x}{π}))}^{(n)} + C_{n}^{1} {3 x}^{2} {(\arctan (\frac{x}{π}))}^{(n - 1)}

+ C_{n}^{2} 6 x {(\arctan (\frac{x}{π}))}^{(n - 2)} + C_{n}^{3} 6 {(\arctan (\frac{x}{π}))}^{(n - 3)}

we have {(\arctan (\frac{x}{π}))}^{(1)} = \frac{1}{π (1 + \frac{x^{2}}{π^{2}})} = \frac{π}{x^{2} + π^{2}}

= \frac{π}{(x - i π) (x + i π)} = π (\frac{1}{x - i π} - \frac{1}{x + i π}) . \frac{1}{2 i π} = \frac{1}{2 i} (\frac{1}{x - i π} - \frac{1}{x + i π})

\Rightarrow {(\arctan (\frac{x}{π}))}^{(m)} = \frac{1}{2 i} (\frac{{(- 1)}^{m - 1} (m - 1)!}{{(x - i π)}^{m}} - \frac{{(- 1)}^{m - 1} (m - 1)!}{{(x + i π)}^{m}})

= \frac{{(- 1)}^{m - 1} (m - 1)!}{2 i} (\frac{{(x + i π)}^{m} - {(x - i π)}^{m}}{{(x^{2} + π^{2})}^{m}})

= \frac{{(- 1)}^{m - 1} (m - 1)! Im {((x + i π))}^{m}}{{(x^{2} + π^{2})}^{m}} \Rightarrow

v^{(n)} (x) = C_{n}^{o} x^{3} . \frac{{{(- 1)}^{n - 1} (n - 1)! Im ((x + i π)))}^{n}}{{(x^{2} + π^{2})}^{n}}

+ {3 C}_{n}^{1} x^{2} . \frac{{(- 1)}^{n - 2} (n - 2)! Im {((x + i π))}^{n - 1}}{{(x^{2} + π^{2})}^{n - 1}}

+ {6 C}_{n}^{2} x . \frac{{(- 1)}^{n - 3} (n - 3)! Im {((x + i π))}^{n - 2}}{{(x^{2} + π^{2})}^{n - 3}}

+ 6 C_{n}^{3} . \frac{{{(- 1)}^{n - 4} (n - 4)! Im ((x + i π)))}^{n - 3}}{{(x^{2} + π^{2})}^{n - 3}}

and f^{(n)} (x) = u^{(n)} (x) + v^{(n)} (x) \Rightarrow

f^{(n)} (1) = u^{(n)} (1) + v^{(n)} (1) \dots .

Commented by mathmax by abdo last updated on 22/Mar/21

if x < 0 we do ch . x = - t \Rightarrow f (x) = f (- t) = {(- t)}^{3} \arctan (\frac{π}{- t})

= t^{3} \arctan (\frac{π}{t}) = t^{3} (\frac{π}{2} - arctant) = - x^{3} (\frac{π}{2} + \arctan (x)) \dots