e-ax-e-x-1-dx-

Question Number 30321 by rahul 19 last updated on 20/Feb/18

\int_{- \infty}^{\infty} \frac{e^{a x}}{e^{x} + 1} d x = ?

Commented by rahul 19 last updated on 20/Feb/18

{what}^{'} s residus theorem ?

Commented by abdo imad last updated on 20/Feb/18

c h . e^{x} = t \Leftrightarrow x = l n t \Rightarrow I = \int_{0}^{\infty} \frac{t^{a}}{t + 1} \frac{d t}{t} = \int_{0}^{\infty} \frac{t^{a - 1}}{1 + t} d t

t h i s i n t e g r a l c o n v e r g e s \Leftrightarrow 0 < a < 1 a n d w e p r o v e b y

r e s i d u s t h e o r e m t h a t I = \frac{π}{s i n (π a)} .

Commented by abdo imad last updated on 20/Feb/18

f o r t h i s t h e o r e m t h e r e a r e m a n y c a s e s l e t t a k e f o r e x a m p l e

F (x) = \frac{p (x)}{q (x)} w i t h p a n d q p o l y n o i a l s / \int_{- \infty}^{+ \infty} \frac{p (x)}{q (x)} d x i s

c o n v e r g e n t a n d q (x) h a v e r o o t s n o t r e a l s {(z_{i})}_{1 ⩽ i ⩽ p} s o

\int_{- \infty}^{+ \infty} \frac{p (x)}{q (x)} d x = 2 i π \sum_{z \in P^{+}} R e s (F, z) w i t h

P^{+} = {z \in C / z p o l e o f F a n d i m (z) > 0} b u t y o u m u s t t a k e

a l o o k i n t h e b o o k o f c o m p l e x a n a l y s i s \dots