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Question Number 44706 by maxmathsup by imad last updated on 03/Oct/18

l e t f_{α} (x) = \frac{c o s (α x)}{1 + x^{2}}

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

3) g i v e \int_{0}^{x} f_{α} (t) d t a t f o r m o f s e r i e

4) d e v e l o p p \int_{0}^{\infty} f_{α} (t) d t a t i n t e g r s e r i e .

Commented by maxmathsup by imad last updated on 04/Oct/18

1) w e h a v e f_{α} (x) = \frac{c o s (α x)}{(x - i) (x + i)} = \frac{c o s (α x)}{2 i} {\frac{1}{x - i} - \frac{1}{x + i}}

= \frac{1}{2 i} {\frac{c o s (α x)}{x - i} - \frac{c o s (α x)}{x + i}} \Rightarrow f^{(n)} (x) = \frac{1}{2 i} {{(\frac{c o s (α x)}{x - i})}^{(n)} - {(\frac{c o s (α x)}{x + i})}^{(n)}} b u t

{(\frac{c o s (α x)}{(x - i)})}^{(n)} = \sum_{k = 0}^{n} C_{n}^{k} {(c o s {(α x)}^{})}^{(k)} {(\frac{1}{x - i})}^{(n - k)} (l e i b n i z f o r m u l a e)

= \sum_{k = 0}^{n} C_{n}^{k} c o s (α x + \frac{k π}{2}) \frac{{(- 1)}^{n - k} (n - k)!}{{(x - i)}^{n - k + 1}} a l s o

{(\frac{c o s (α x)}{x + i})}^{(n)} = \sum_{k = 0}^{n} C_{n}^{k} c o s (α x + \frac{π}{2}) \frac{{(- 1)}^{n - k} (n - k)!}{{(x + i)}^{n - k + 1}} \Rightarrow

f^{(n)} (x) = \frac{1}{2 i} \sum_{k = 0}^{n} C_{n}^{k} {(- 1)}^{n - k} (n - k)! c o s (α x + \frac{k π}{2}) {\frac{1}{{(x - i)}^{n - k + 1}} - \frac{1}{{(x + i)}^{n - k + 1}}}

f^{(n)} (0) = \frac{1}{2 i} \sum_{k = 0}^{n} {(- 1)}^{n - k} \frac{n!}{k!} c o s (\frac{k π}{2}) {i^{n - k + 1} - {(- i)}^{n - k + 1}}

= \sum_{k = 0}^{n} {(- 1)}^{n - k} \frac{n!}{k!} c o s (\frac{k π}{2}) s i n (n - k + 1) \frac{π}{2}

Commented by maxmathsup by imad last updated on 05/Oct/18

2) w e h a v e f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n}

= \sum_{n = 0}^{\infty} (\sum_{k = 0}^{n} \frac{{(- 1)}^{n - k}}{k!} c o s (\frac{k π}{2}) s i n (n - k + 1) \frac{π}{2}) x^{n}

= \sum_{n = 0}^{\infty} (\sum_{k = 0}^{n} \frac{{(- 1)}^{n - k}}{k!} c o s (\frac{k π}{2}) c o s ((n - k) \frac{π}{2}) x^{n} .

Commented by maxmathsup by imad last updated on 05/Oct/18

3) \int_{0}^{x} f_{α} (t) d t = \sum_{n = 0}^{\infty} (\sum_{k = 0}^{n} \frac{{(- 1)}^{n - k}}{k!} c o s (\frac{k π}{2}) c o s ((n - k) \frac{π}{2}) \frac{x^{n + 1}}{n + 1} .

Commented by maxmathsup by imad last updated on 05/Oct/18

4) l e t c a l c u l a t e f i r s t \int_{0}^{\infty} f_{α} (t) d t = I_{α}

I_{α} = \int_{0}^{\infty} \frac{c o s (α t)}{1 + t^{2}} d t \Rightarrow 2 I_{α} = \int_{- \infty}^{+ \infty} \frac{c o s (α t)}{1 + t^{2}} d t = R e (\int_{- \infty}^{+ \infty} \frac{e^{i α t}}{1 + t^{2}} d t)

l e t φ (z) = \frac{e^{i α z}}{1 + z^{2}} t h e p o l e s o f φ a r e i a n d - i r e s i d u s t h e o r e m g i v e

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i) = 2 i π \frac{e^{- α}}{2 i} = π e^{- α} \Rightarrow I_{α} = \frac{π}{2} e^{- α}

\Rightarrow \int_{0}^{\infty} f_{α} (t) d t = \frac{π}{2} e^{- α} = \frac{π}{2} \sum_{n = 0}^{\infty} \frac{{(- α)}^{n}}{n!} = \frac{π}{2} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n!} α^{n} .