calculate-f-x-cos-x-1-t-2-1-t-2-dt-with-x-0-

Question Number 70871 by Abdo msup. last updated on 09/Oct/19

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

Commented by mathmax by abdo last updated on 10/Oct/19

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

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

\int_{- \infty}^{+ \infty} W (z) d z = 2 i π R e s (W, i) = 2 i π e^{i x} \times \frac{e^{- i x}}{2 i} = π \Rightarrow

f (x) = π \forall x