calculate-f-t-cos-tx-1-x-2-dx-

Question Number 37287 by math khazana by abdo last updated on 11/Jun/18

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

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

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

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

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

= π e^{- t} w i t h t ⩾ 0 .