find-0-dx-1-x-2-n-with-n-integr-and-n-1-

Question Number 31296 by abdo imad last updated on 05/Mar/18

f i n d \int_{0}^{+ \infty} \frac{d x}{{(1 + x^{2})}^{n}} w i t h n i n t e g r a n d n ⩾ 1 .

Commented by NECx last updated on 07/Mar/18

t h a t s t h e g r e a t e s t l i e o f t h e c e n t u r y .

Commented by abdo imad last updated on 06/Mar/18

l e t p u t I_{n} = \int_{0}^{\infty} \frac{d x}{{(1 + x^{2})}^{n}} \Rightarrow I_{n} = \frac{1}{2} \int_{- \infty}^{+ \infty} \frac{d x}{{(1 + x^{2})}^{n}}

l e t p u t φ (z) = \frac{1}{{(1 + z^{2})}^{n}} w e h a v e φ (z) = \frac{1}{{(z - i)}^{n} {(z + i)}^{n}} s o

t h e p o l e s o f f a r e i a n d - i (p o l e s w i t h o r d r e n)

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i) b u t

R e s (φ, i) = {l i m}_{z \to i} \frac{1}{(n - 1)!} {({(z - i)}^{n} f (z))}^{(n - 1)}

= {l i m}_{z \to i} \frac{1}{(n - 1)!} {(\frac{1}{{(z + i)}^{n}})}^{(n - 1)} f i r s t l e t f i n d

{({(z + i)}^{p})}^{(k)} w i t h k ⩽ p

{{(z + i)}^{p})}^{(1)} = p {(z + i)}^{p - 1}, {({(z + i)}^{p})}^{(2)} = p (p - 1) {(z + i)}^{p - 2} \Rightarrow

{({(z + i)}^{p})}^{(k)} = p (p - 1) \dots (p - k + 1) {(z + i)}^{p - k} a n d

f o r p = - n a n d k = n - 1 w e g e t

{({(z + i)}^{- n})}^{(n - 1)} = (- n) (- n - 1) \dots . (- n - n + 1 + 1) {(z + i)}^{- n - n + 1}

= {(- 1)}^{n - 1} n (n + 1) (n + 2) \dots . (2 n - 2) {(z + i)}^{- 2 n + 1}

R e s (φ, i) = \frac{1}{(n - 1)!} {(- 1)}^{n - 1} n (n + 1) (n + 2) \dots (2 n - 2) {(2 i)}^{- 2 n + 1}

= \frac{{(- 1)}^{n - 1}}{2^{2 n - 1} (n - 1)! i^{2 n - 1}} n (n + 1) (n + 2) \dots . (2 n - 2)

= \frac{- i}{2^{2 n - 1}} \frac{1}{(n - 1)!} n (n + 1) (n + 2) \dots (2 n - 2) \Rightarrow

I_{n} = \frac{1}{2} 2 i π \frac{- i}{2^{2 n - 1}} \frac{1}{(n - 1)!} n (n + 1) (n + 2) \dots (2 n - 2)

I_{n} = \frac{π}{2^{2 n - 1}} n (n + 1) (n + 2) \dots . (2 n - 2) = \frac{π (2 n - 2)!}{{((n - 1)!)}^{2} 2^{2 n - 1}} .

Commented by NECx last updated on 07/Mar/18

w o w \dots . . P r o f A b d o I m a d, I^{'} m s o

a m a s e d b y t h e q u e s t i o n s y o u

s o l v e . T h a n k s f o r y o u h e l p o n t h e

p l a t f o r m .

Commented by NECx last updated on 07/Mar/18

I f I m a y a s k, w h i c h a s p e c t o f

i n t e g r a t i o n i s t h i s ?

Commented by NECx last updated on 07/Mar/18

t h i s i s b e c a u s e I a l w a y s n o t i c e

t h a t y o u a l w a y s a p p l y c o m p l e x

v a l u e s t o y o u r i n t e g r a l s .

Commented by prof Abdo imad last updated on 07/Mar/18

i t s o n l y i n t e g r a t i o n b y r e s i d u s t h e o r e m \dots .

Commented by prof Abdo imad last updated on 07/Mar/18

I a m g i v e n a l l m y t i m e t o i n t e g r a t i o n b e c a u s e

i t s c o n t a i n a l l c o m s e p t s o f a n a l y s i s \dots

Commented by NECx last updated on 07/Mar/18

w o w \dots . i^{'} m m o s t g r a t e f u l

Commented by NECx last updated on 07/Mar/18

I^{'} v e h e a r d o f t h e t h e o r e m : c a u c h y

r e s i d u e t h e o r e m . I t h a s t o d o w i t h

c o m l e x a n a l y s i s . T h o u g h i t s n o t

i n C l a s s X I o r X I I s y l l a b u s . u

Commented by rahul 19 last updated on 07/Mar/18

s o u r a l s o a p r o f ?

Commented by rahul 19 last updated on 07/Mar/18

t h e n ?