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

$$calculatef\left(t\right)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{cos\left(tx\right)}{1+x^2}dx$$

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

$$f\left(t\right)=Re\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{itx}}{1+x^2}dx\right)let$$$$\phi \left(z\right)=\frac{e^{itz}}{1+z^2}thepolesof\phi areiand-i$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi Res(\phi ,i)=2i\pi \frac{e^{it\left(i\right)}}{2i}$$$$=\pi e^{-t}witht⩾0.$$

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