calculate-f-cos-2x-1-ax-2-dx-with-a-gt-0-2-find-the-value-of-cos-2x-1-3x-2-dx-

Question Number 37288 by math khazana by abdo last updated on 11/Jun/18

c a l c u l a t e f (α) = \int_{- \infty}^{+ \infty} \frac{c o s (2 x)}{1 + {a x}^{2}} d x w i t h a > 0

2) f i n d t h e v a l u e o f \int_{- \infty}^{+ \infty} \frac{c o s (2 x)}{1 + 3 x^{2}} d x .

Commented by prof Abdo imad last updated on 16/Jun/18

1) f (a) = R e (\int_{- \infty}^{+ \infty} \frac{e^{i 2 x}}{1 + {a x}^{2}} d x) l e t

φ (z) = \frac{e^{2 i z}}{1 + {a z}^{2}}

φ (z) = \frac{e^{2 i z}}{{(\sqrt{a} z)}^{2} - i^{2}} = \frac{e^{2 i z}}{(\sqrt{a} z - i) (\sqrt{a} z + i)}

= \frac{e^{2 i z}}{a (z - \frac{i}{\sqrt{a}}) (z + \frac{i}{\sqrt{a}})} s o t h e p o l e s o f φ a r e \frac{i}{\sqrt{a}}

a n d \frac{- i}{\sqrt{a}}

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

R e s (φ, \frac{i}{\sqrt{a}}) = \frac{e^{2 i \frac{i}{\sqrt{a}}}}{a \frac{2 i}{\sqrt{a}}} = \frac{e^{\frac{- 2}{\sqrt{a}}}}{2 i \sqrt{a}} \Rightarrow

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

f (a) = \frac{π}{\sqrt{a}} e^{- \frac{2}{\sqrt{a}}}

2) \int_{- \infty}^{+ \infty} \frac{c o s (2 x)}{1 + 3 x^{2}} d x = f (3)

= \frac{π}{\sqrt{3}} e^{- \frac{2}{\sqrt{3}}} .