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Question Number 29038 by abdo imad last updated on 03/Feb/18

$$find{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{cos\left(at\right)}{1+t^4}dt.$$

Commented by abdo imad last updated on 11/Feb/18

$$letputI\left(a\right)={\underset{-\mathrm{\infty }}{\int }}^{+\mathrm{\infty }}\frac{cos\left(at\right)}{1+t^4}dt=Re\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{iat}}{1+t^4}dt\right)let$$$$introducethecomplexfunction\phi \left(z\right)=\frac{e^{iaz}}{1+z^4}$$$$polesof\phi ?z^4=-1\iff z^4=e^{i\pi }ifz=r{e}^{i\theta }\Rightarrow$$$$r=1and4\theta =(2k+1)\pi \Rightarrow \theta =(2k+1)\frac{\pi }{4}sothepolesof$$$$\phi are{z}_k=e^{i(2k+1)\frac{\pi }{4}}andk\in \left[[o,3]\right]$$$$z_0=e^{i\frac{\pi }{4}},z_1=e^{i\frac{3\pi }{4}},z_2=e^{i\frac{5\pi }{4}},z_3=e^{k\frac{7\pi }{4}}$$$$\phi \left(z\right)=\frac{e^{iaz}}{(z-z_0)(z-z_1)(z-z_2)(z-z_3)}$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi (Re(\phi ,z_0)+Re(\phi ,z_1))$$$$Res(\phi ,z_0)=\frac{e^{{iaz}_0}}{4z_{0^{}}^3}=-\frac{1}{4}{z}_0{e}^{{iaz}_0}$$$$Res(\phi ,z_1)=\frac{e^{{iaz}_1}}{4z_1^3}=-\frac{1}{4}{z}_1{e}^{{iaz}_1}but{z}_1=z_0^{-}$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi (-\frac{1}{4})(z_0{e}^{{iaz}_0}+z_0^{-}{e}^{{iaz}_0^{-}})$$$$=-\frac{i\pi }{2}(z_0{e}^{{iaz}_0}+z_0^{-}{e}^{{iaz}_9^{-}})....becontinued....$$

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