let-x-arctan-2x-1-x-2-1-calculate-n-x-2-calculate-n-0-anddevelpp-at-integr-serie-

Question Number 53961 by maxmathsup by imad last updated on 27/Jan/19

l e t φ (x) = \frac{a r c t a n (2 x)}{1 - x^{2}}

1) c a l c u l a t e φ^{(n)} (x)

2) c a l c u l a t e φ^{(n)} (0) a n d d e v e l p p φ a t i n t e g r s e r i e

Commented by maxmathsup by imad last updated on 06/Feb/19

1) w e h a v e φ (x) = \frac{1}{2} a r c t a n (2 x) {\frac{1}{1 - x} + \frac{1}{1 + x}) = \frac{1}{2} \frac{a r c t a n (2 x)}{1 - x} + \frac{1}{2} \frac{a r c t a n (2 x)}{1 + x}

= W (x) - H (x) w i t h W (x) = \frac{1}{2} \frac{a r c t a n (2 x)}{x + 1} a n d H (x) = \frac{1}{2} \frac{a r c t a n (2 x)}{x - 1}

\Rightarrow φ^{(n)} (x) = W^{(n)} (x) - H^{(n)} (x) l e i b n i z f o r m u l a g i v e

W^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(a r c t a n (2 x))}^{(k)} {(\frac{1}{x + 1})}^{(n - k)} b u t

{(\frac{1}{x + 1})}^{(n - k)} = \frac{{(- 1)}^{n - k} (n - k)!}{{(x + 1)}^{n - k + 1}}

w e h a v e {(a r c t a n (2 x))}^{^{'}} = \frac{2}{1 + 4 x^{2}} \Rightarrow {(a r c t a n (2 x))}^{(k)} = 2 {(\frac{1}{4 x^{2} + 1})}^{(k - 1)}

= - i {\frac{1}{2 x - i} - \frac{1}{2 x + i}}^{(k - 1)} = i {\frac{1}{2 (x + \frac{i}{2})} - \frac{1}{2 (x - \frac{i}{2})}}^{(k - 1)}

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

= \frac{i}{2} {(- 1)}^{k - 1} (k - 1)! {\frac{{(x - \frac{i}{2})}^{k} - {(x + \frac{i}{2})}^{k}}{{(x^{2} + \frac{1}{4})}^{k}}} \Rightarrow

W^{(n)} (x) = a r c t a n (2 x) \frac{{(- 1)}^{n} n!}{{(x + 1)}^{n + 1}} + \sum_{k = 1}^{n} C_{n}^{k} \frac{i}{2} {(- 1)}^{k - 1} (k - 1)! {\frac{{(x - \frac{i}{2})}^{k} - {(x + \frac{i}{2})}^{k}}{{(x^{2} + \frac{1}{4})}^{k}}} \frac{{(- 1)}^{n - k} (n - k)!}{{(x + 1)}^{n - k + 1}} .

= \frac{{(- 1)}^{n} n!}{{(x + 1)}^{n + 1}} a r c t a n (2 x) - \sum_{k = 1}^{n} \frac{n!}{k} {(- 1)}^{k} (\frac{I m {(x + \frac{i}{2})}^{k}}{{(x^{2} + \frac{1}{4})}^{k}}) \frac{1}{{(x + 1)}^{n - k + 1}} .

Commented by maxmathsup by imad last updated on 06/Feb/19

a l s o w e h a v e

H^{(n)} (x) = \frac{{(- 1)}^{n} n!}{(x - 1)!} a r c t a n (2 x) - \sum_{k = 1}^{n} n! \frac{{(- 1)}^{k}}{k} \frac{I m {(x + \frac{i}{2})}^{k}}{{(x^{2} + \frac{1}{4})}^{k} {(x - 1)}^{n - k + 1}} .

φ^{(n)} (0) = W^{(n)} (0) - H^{(n)} (0)

= n! \sum_{k = 1}^{n} \frac{{(- 1)}^{k - 1}}{k} \frac{I m (\frac{i^{k}}{2^{k}})}{{(\frac{1}{4})}^{k}} - (n! \sum_{k = 1}^{n} \frac{{(- 1)}^{k - 1}}{k} \frac{I m (\frac{i^{k}}{2^{k}})}{{(\frac{1}{4})}^{k}} {(- 1)}^{n - k + 1}

= n! {\sum_{k = 1}^{n} \frac{{(- 1)}^{k - 1}}{k} 4^{- k} \frac{1}{2^{k}} s i n (\frac{k π}{2}) - \sum_{k = 1}^{n} {(- 1)}^{n} 4^{- k} \frac{1}{2^{k}} s i n (\frac{k π}{2})} .