1-ix-1-ix-n-1-ix-1-ix-m-1-1-x-2-dx-

Question Number 151983 by puissant last updated on 24/Aug/21

\int_{- \infty}^{+ \infty} {(\frac{1 - i x}{1 + i x})}^{n} {(\frac{1 + i x}{1 - i x})}^{m} \frac{1}{1 + x^{2}} d x

Answered by Olaf_Thorendsen last updated on 24/Aug/21

I = \int_{- \infty}^{+ \infty} {(\frac{1 - i x}{1 + i x})}^{n} {(\frac{1 + i x}{1 - i x})}^{m} \frac{1}{1 + x^{2}} d x

∙ m = n, I = π (trivial)

∙ m \neq n :

I = \int_{- \infty}^{+ \infty} \frac{e^{i n \arctan (- x)}}{e^{i n \arctan (+ x)}} . \frac{e^{i m \arctan (+ x)}}{e^{i m \arctan (- x)}} . \frac{1}{1 + x^{2}} d x

I = \int_{- \infty}^{+ \infty} \frac{e^{2 i (m - n) \arctan x}}{1 + x^{2}} d x

I = {[\frac{e^{2 i (m - n) \arctan x}}{2 i (m - n)}]}_{- \infty}^{+ \infty}

I = \frac{e^{i π (m - n)} - e^{- i π (m - n)}}{2 i (m - n)}

I = \frac{\sin (π (m - n))}{m - n} = \frac{0}{m - n} = 0

My result is very strange .

You should verify the calculous .

I^{'} m not sure {it}^{'} s the good way .

Commented by puissant last updated on 24/Aug/21

m e r c i b e a u c o u p M r

Commented by puissant last updated on 24/Aug/21

\int_{- \infty}^{+ \infty} {(\frac{1 - i x}{1 + i x})}^{n} {(\frac{1 + i x}{1 - i x})}^{m} \frac{1}{1 + x^{2}} d x

= \int_{- \infty}^{+ \infty} {(\frac{1 + i x}{1 - i x})}^{m - n} \frac{1}{1 + x^{2}} d x

= \int_{- \frac{π}{2}}^{+ \frac{π}{2}} {(\frac{1 + i t a n t}{1 - i t a n t})}^{m - n} d t = \int_{- \frac{π}{2}}^{+ \frac{π}{2}} {(\frac{c o s t + i s i n t}{c o s t - i s i n t})}^{m - n} d t

= \int_{- \frac{π}{2}}^{+ \frac{π}{2}} e^{2 i t (m - n)} d t = {[\frac{e^{2 i t (m - n)}}{2 i (m - n)}]}_{- \frac{π}{2}}^{+ \frac{π}{2}}

= \frac{e^{i π (m - n)} - e^{- i π (m - n)}}{2 i (m - n)}

= \frac{s i n (π (m - n))}{(m - n)} = 0 . .

E n f a i t v o i c i c e q u e j^{'} a i f a i t M r m a i s

j e v o u l a i s c o n f i r m e r m o n r e s u l t a t . .

M e r c i p o u r v o t r e t e m p s . .

C o r d i a l e m e n t . .

Commented by Olaf_Thorendsen last updated on 24/Aug/21

Excellent!

Cette integrale fait peur au depart mais

elle se calcule en 4 ou 5 lignes .

C^{'} est souvent trompeur .