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

$$letafromRfind{F}_a\left(t\right)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{cos\left(tx\right)}{a^2+x^2}dx$$$$2)calculate{F}_2\left(3\right)and{F}_3\left(2\right)$$

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

$$wehave{F}_a\left(t\right)=Re\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{itx}}{x^2+a^2}dx\right)$$$$let\phi \left(z\right)=\frac{e^{itz}}{z^2+a^2}thepolesof\phi areiaand-ia$$$$case1a>0$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi Res(\phi ,ia)=2i\pi \frac{e^{it\left(ia\right)}}{2ia}$$$$=\frac{\pi }{a}{e}^{-at}\Rightarrow F_a\left(t\right)=\frac{\pi }{a}{e}^{-at}$$$$case2a<0$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi Res(\phi ,-ia)=2i\pi \frac{e^{it(-ia)}}{-2ia}$$$$=-\frac{\pi }{a}{e}^{at}\Rightarrow F_a\left(t\right)=-\frac{\pi }{a}{e}^{at}$$$$2)F_2\left(3\right)=\frac{\pi }{2}{e}^{-6}and{F}_3\left(2\right)=\frac{\pi }{3}{e}^{-6}$$

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