let-a-from-R-find-F-a-t-cos-tx-a-2-x-2-dx-2-calculate-F-2-3-and-F-3-2-

Question Number 38466 by maxmathsup by imad last updated on 25/Jun/18

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

2) c a l c u l a t e F_{2} (3) a n d F_{3} (2)

Commented by math khazana by abdo last updated on 27/Jun/18

w e h a v e F_{a} (t) = R e (\int_{- \infty}^{+ \infty} \frac{e^{i t x}}{x^{2} + a^{2}} d x)

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

c a s e 1 a > 0

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

= \frac{π}{a} e^{- a t} \Rightarrow F_{a} (t) = \frac{π}{a} e^{- a t}

c a s e 2 a < 0

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

= - \frac{π}{a} e^{a t} \Rightarrow F_{a} (t) = - \frac{π}{a} e^{a t}

2) F_{2} (3) = \frac{π}{2} e^{- 6} a n d F_{3} (2) = \frac{π}{3} e^{- 6}