find-the-value-of-cos-ax-1-x-x-2-dx-

Question Number 33204 by prof Abdo imad last updated on 12/Apr/18

f i n d t h e v a l u e o f \int_{- \infty}^{+ \infty} \frac{c o s (a x)}{1 + x + x^{2}} d x .

Commented by abdo imad last updated on 13/Apr/18

l e t p u t f (a) = \int_{- \infty}^{+ \infty} \frac{c o s (a x)}{1 + x + x^{2}} d x

f (a) = R e (\int_{- \infty}^{+ \infty} \frac{e^{i a x}}{1 + x + x^{2}} d x) l e t c o n s i d e r t h e c o m p l e x

f u n c t i o n φ (z) = \frac{e^{i a z}}{1 + z + z^{2}} t h e p o l e s o f φ a r e j a n d \bar{j}

j = e^{i \frac{2 π}{3}} a n d φ (z) = \frac{e^{i a z}}{(z - j) (z - \bar{j})}

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

R e s (φ, j) = \frac{e^{i a j}}{j - \bar{j}} = \frac{e^{i a (- \frac{1}{2} + i \frac{\sqrt{3}}{2})}}{2 i I m (j)} = \frac{e^{- \frac{i a}{2}} . e^{- a \frac{\sqrt{3}}{2}}}{2 i \frac{\sqrt{3}}{2}}

= \frac{e^{- a \frac{\sqrt{3}}{2}}}{i \sqrt{3}} (c o s (\frac{a}{2}) - i s i n (\frac{a}{2})) \Rightarrow

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{e^{- a \frac{\sqrt{3}}{2}}}{i \sqrt{3}} (c o s (\frac{a}{2}) - i s i n (\frac{a}{2})) \Rightarrow

f (a) = \frac{2 π}{\sqrt{3}} c o s (\frac{a}{2}) . e^{- a \frac{\sqrt{3}}{2}} .