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Question Number 33204 by prof Abdo imad last updated on 12/Apr/18

$$findthevalueof{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{cos\left(ax\right)}{1+x+x^2}dx.$$

Commented by abdo imad last updated on 13/Apr/18

$$letputf\left(a\right)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{cos\left(ax\right)}{1+x+x^2}dx$$$$f\left(a\right)=Re\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{iax}}{1+x+x^2}dx\right)letconsiderthecomplex$$$$function\phi \left(z\right)=\frac{e^{iaz}}{1+z+z^2}thepolesof\phi arejand\stackrel{-}{j}$$$$j=e^{i\frac{2\pi }{3}}and\phi \left(z\right)=\frac{e^{iaz}}{(z-j)(z-\stackrel{-}{j})}$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi Res(\phi ,j)$$$$Res(\phi ,j)=\frac{e^{iaj}}{j-\stackrel{-}{j}}=\frac{e^{ia(-\frac{1}{2}+i\frac{\sqrt{3}}{2})}}{2iIm\left(j\right)}=\frac{e^{-\frac{ia}{2}}.e^{-a\frac{\sqrt{3}}{2}}}{2i\frac{\sqrt{3}}{2}}$$$$=\frac{e^{-a\frac{\sqrt{3}}{2}}}{i\sqrt{3}}(cos\left(\frac{a}{2}\right)-isin\left(\frac{a}{2}\right))\Rightarrow$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi \frac{e^{-a\frac{\sqrt{3}}{2}}}{i\sqrt{3}}(cos\left(\frac{a}{2}\right)-isin\left(\frac{a}{2}\right))\Rightarrow$$$$f\left(a\right)=\frac{2\pi }{\sqrt{3}}cos\left(\frac{a}{2}\right).e^{-a\frac{\sqrt{3}}{2}}.$$$$$$

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