Question Number 65131 by turbo msup by abdo last updated on 25/Jul/19
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{4}{i}\right)^{\mathrm{3}} } \\ $$
Commented by mathmax by abdo last updated on 26/Jul/19
$${let}\:{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{4}{i}\right)^{\mathrm{3}} }\:\Rightarrow\mathrm{2}{I}\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{4}{i}\right)^{\mathrm{3}} } \\ $$$$\:{let}\:\varphi\left({z}\right)\:=\frac{\mathrm{1}}{\left({z}^{\mathrm{4}} −\mathrm{4}{i}\right)^{\mathrm{3}} }\:\:{poles}\:{of}\:\varphi?\:\:{first}\:{let}\:{solve}\:{z}^{\mathrm{4}} \:=\mathrm{4}{i} \\ $$$${z}\:={r}\:{e}^{{i}\theta} \:\Rightarrow\:{r}^{\mathrm{4}} \:{e}^{{i}\mathrm{4}\theta} \:=\mathrm{4}\:{e}^{\frac{{i}\pi}{\mathrm{2}}} \:\Rightarrow{r}^{\mathrm{4}} \:=\mathrm{4}\:\:{and}\:\mathrm{4}\theta\:=\frac{\pi}{\mathrm{2}}\:+\mathrm{2}{k}\pi\:\Rightarrow{r}\:=\mathrm{4}^{\frac{\mathrm{1}}{\mathrm{4}}} \\ $$$$=\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \:=\sqrt{\mathrm{2}}\:{and}\:\:\theta\:=\frac{\pi}{\mathrm{8}}\:+\frac{{k}\pi}{\mathrm{2}}\:\Rightarrow\mathrm{4}{i}\:=\left(\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{4}} \:\Rightarrow \\ $$$${z}^{\mathrm{4}} −\mathrm{4}{i}\:={z}^{\mathrm{4}} −\left(\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{4}} \:=\left({z}^{\mathrm{2}} \:−\left(\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{2}} \right)\left({z}^{\mathrm{2}} \:+\left(\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{2}} \right) \\ $$$$=\left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)\left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)\left({z}^{\mathrm{2}} −\left(\sqrt{\mathrm{2}}{i}\:{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{2}} \right) \\ $$$$=\left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)\left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)\left({z}^{\mathrm{2}} −\left(\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)^{\mathrm{2}} \right) \\ $$$$\left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)\left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)\left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)\left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)\:\Rightarrow \\ $$$$\varphi\left({z}\right)\:=\frac{\mathrm{1}}{\left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} \left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} \left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} \left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} } \\ $$$${the}\:{poles}\:{of}\:\varphi\:{are}\:\overset{−} {+}\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \:\:{and}\:\overset{−} {+}\:{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \:\:\:\left({ordre}\:\mathrm{3}\right) \\ $$$${residus}\:{theorem}\:{give}\:\int_{−\infty} ^{+\infty} \:\:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:\left\{{Res}\left(\varphi,\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)+{Res}\left(\varphi,\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)\right\} \\ $$$${Res}\left(\varphi,\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)\:={lim}_{{z}\rightarrow\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} } \:\:\:\frac{\mathrm{1}}{\left(\mathrm{3}−\mathrm{1}\right)!}\left\{\:\left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} \varphi\left({z}\right)\right\}^{\left(\mathrm{2}\right)} \\ $$$$={lim}_{{z}\rightarrow\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} } \:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\left\{\frac{\mathrm{1}}{\left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} \left({z}−\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} \left({z}+\sqrt{\mathrm{2}}{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{8}}} \right)^{\mathrm{3}} }\right\}^{\left(\mathrm{2}\right)} \\ $$$$…{be}\:{continued}…. \\ $$