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Question Number 108502 by mnjuly1970 last updated on 17/Aug/20

Commented by mnjuly1970 last updated on 17/Aug/20

$$\mathrm{♣}pleaseproveaboveintegrals$$$$equalities.\mathrm{♣}$$

Answered by mathmax by abdo last updated on 17/Aug/20

$$1){\mathrm{A}}_{\mathrm{p}}={\int }_0^{\frac{\pi }{2}}\sin \left(\mathrm{ptanx}\right)\mathrm{tanx}\mathrm{dx}\mathrm{changement}\mathrm{tanx}=\mathrm{t}\mathrm{give}$$$${\mathrm{A}}_{\mathrm{p}}={\int }_0^{\mathrm{\infty }}\sin \left(\mathrm{pt}\right)\mathrm{t}\frac{\mathrm{dt}}{1+{\mathrm{t}}^2}={\int }_0^{\mathrm{\infty }}\frac{\mathrm{tsin}\left(\mathrm{pt}\right)}{1+{\mathrm{t}}^2}\mathrm{dt}$$$$=\frac{1}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{\mathrm{tsin}\left(\mathrm{pt}\right)}{{\mathrm{t}}^2+1}\mathrm{dt}=\frac{1}{2}\mathrm{Im}\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{{\mathrm{te}}^{\mathrm{ipt}}}{{\mathrm{t}}^2+1}\mathrm{dt}\right)\mathrm{let}$$$$\phi \left(\mathrm{z}\right)=\frac{\mathrm{z}{\mathrm{e}}^{\mathrm{ipz}}}{{\mathrm{z}}^2+1}\Rightarrow \phi \left(\mathrm{z}\right)=\frac{\mathrm{z}{\mathrm{e}}^{\mathrm{ipz}}}{(\mathrm{z}-\mathrm{i})(\mathrm{z}+\mathrm{i})}\mathrm{residus}\mathrm{theorem}\mathrm{give}$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(\mathrm{z}\right)\mathrm{dz}=2\mathrm{i}\pi \mathrm{Res}(\phi ,\mathrm{i})=2\mathrm{i}\pi .\frac{\mathrm{i}{\mathrm{e}}^{-\mathrm{p}}}{2\mathrm{i}}=\mathrm{i}\pi {\mathrm{e}}^{-\mathrm{p}}\Rightarrow$$$$★{\mathrm{A}}_{\mathrm{p}}=\frac{\pi }{2}{\mathrm{e}}^{-\mathrm{p}}★$$

Commented by mnjuly1970 last updated on 17/Aug/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Commented by mathmax by abdo last updated on 17/Aug/20

$$\mathrm{you}\mathrm{are}\mathrm{welcome}$$

Commented by mnjuly1970 last updated on 17/Aug/20

$$\mathrm{the}\mathrm{god}\mathrm{bless}\mathrm{you}\mathrm{and}\mathrm{keep}\mathrm{you}\mathrm{my}$$$$\mathrm{friend}\mathrm{and}\mathrm{my}\mathrm{masster}..{\left(\mathrm{thank}\mathrm{you}\right)}^{\mathrm{\infty }}$$$$$$

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