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Question Number 55270 by maxmathsup by imad last updated on 20/Feb/19

l e t f (x) = x^{n} a r c t a n (x^{2}) w i t h n i n t e g r n a t u r a l

1) c a l c u l a t e f^{(n)} (x) a n d f^{(n)} (0)

2) d e v e l o p p f a t i n t e g r s e r i e .

Commented by maxmathsup by imad last updated on 02/Mar/19

1) l e i b n i z f o r m u l a g i v e f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(x^{n})}^{(k)} {(a r c t a n (x^{2}))}^{(n - k)}

b u t {(x^{n})}^{(k)} = n (n - 1) \dots (n - k + 1) x^{n - k} = \frac{n!}{(n - k)!} x^{n - k} i f k ⩽ n \Rightarrow

f^{(n)} (x) = \sum_{k = 0}^{n} \frac{n!}{k! (n - k)!} \frac{n!}{(n - k)!} x^{n - k} {a r c t a n (x^{2})}^{) n - k)}

= \sum_{k = 0}^{n} \frac{{(n!)}^{2}}{k! {(n - k)!}^{2}} {a r c t a n (x^{2})}^{(n - k)} l e t f i n d {(a r c t a n (x^{2}))}^{(m)}

w e h a v e {a r c t a n (x^{2})}^{(1)} = \frac{2 x}{1 + x^{4}} = \frac{2 x}{(x^{2} - i) (x^{2} + i)} = \frac{2 x}{(x - \sqrt{i}) (x + \sqrt{i}) (x - \sqrt{- i}) (x + \sqrt{- i})}

= \frac{2 x}{(x - e^{\frac{i π}{4}}) (x + e^{\frac{i π}{4}}) (x - e^{- \frac{i π}{4}}) (x + e^{- \frac{i π}{4}})} = F (x) = \frac{a}{x - e^{\frac{i π}{4}}} + \frac{b}{x + e^{\frac{i π}{4}}} + \frac{c}{x - e^{- \frac{i π}{4}}} + \frac{d}{x + e^{- \frac{i π}{4}}}

a = \frac{2 e^{\frac{i π}{4}}}{2 e^{\frac{i π}{4}} (2 i s i n (\frac{π}{4})) (2 c o s (\frac{π}{4}))} = \frac{1}{4 i \frac{1}{2}} = \frac{1}{2 i}

b = \frac{- 2 e^{\frac{i π}{4}}}{(- 2 e^{\frac{i π}{4}}) (- 2 c o s (\frac{π}{4})) (- 2 i s i n (\frac{π}{4}))} = \frac{1}{2 i}

c = \frac{2 e^{- \frac{i π}{4}}}{(- 2 i s i n (\frac{π}{4})) (2 c o s (\frac{π}{4})) (2 e^{- \frac{i π}{4}})} = - \frac{1}{2 i}

d = \frac{- 2 e^{- \frac{i π}{4}}}{(- 2 c o s (\frac{π}{4})) (2 i s i n (\frac{π}{4})) (- 2 e^{- \frac{i π}{4}})} = - \frac{1}{2 i} \Rightarrow

{(a r c t a n (x^{2}))}^{(1)} = \frac{1}{2 i} {\frac{1}{x - e^{\frac{i π}{4}}} + \frac{1}{x + e^{\frac{i π}{4}}} - \frac{1}{x - e^{- \frac{i π}{4}}} - \frac{1}{x + e^{- \frac{i π}{4}}}} \Rightarrow

{(a r c t a n (x^{2}))}^{(m)} = \frac{1}{2 i} {(- 1)}^{m - 1} (m - 1)! {\frac{1}{{(x - e^{\frac{i π}{4}})}^{m}} + \frac{1}{{(x + e^{\frac{i π}{4}})}^{m}} - \frac{1}{{(x - e^{- \frac{i π}{4}})}^{m}} - \frac{1}{(x + e^{- \frac{i π}{4}})}} \Rightarrow

f^{(n)} (x) = \frac{1}{2 i} \sum_{k = 0}^{n} \frac{{(n!)}^{2}}{k! {(n - k)!}^{2}} {(- 1)}^{n - k - 1} (n - k - 1)! {\frac{1}{{(x - e^{\frac{i π}{4}})}^{n - k}} + \frac{1}{{(x + e^{\frac{i π}{4}})}^{n - k}} - \frac{1}{{(x - e^{- \frac{i π}{4}})}^{n - k}} - \frac{1}{{(x + e^{- \frac{i π}{4}})}^{n - k}}}