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Question Number 75986 by Ajao yinka last updated on 21/Dec/19

Commented by abdomathmax last updated on 24/Dec/19

$$letf\left(t\right)={\int }_0^{\mathrm{\infty }}\frac{arctan\left(xt\right)cos\left(nx\right)}{x}dxwitht⩾0$$$$f^{{}^{\prime }}\left(t\right)={\int }_0^{\mathrm{\infty }}\frac{x}{1+x^2{t}^2}\frac{cos\left(nx\right)}{x}dx={\int }_0^{\mathrm{\infty }}\frac{cos\left(nx\right)}{x^2{t}^2+1}dx$$$${=}_{xt=u}{\int }_0^{\mathrm{\infty }}\frac{cos\left(n\frac{u}{t}\right)}{u^2+1}\frac{du}{t}$$$$=\frac{1}{2t}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{cos\left(\frac{nu}{t}\right)}{u^2+1}du=\frac{1}{2t}Re\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{i\left(\frac{nu}{t}\right)}}{u^2+1}du\right)$$$$letW\left(z\right)=\frac{e^{i\left(\frac{nz}{t}\right)}}{z^2+1}wehsve?$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}W\left(z\right)dz=2i\pi Res(W,i)=2i\pi \times \frac{e^{-\frac{n}{t}}}{2i}$$$$=\pi e^{-\frac{n}{t}}\Rightarrow f^{{}^{\prime }}\left(t\right)=\frac{\pi }{2t}{e}^{-\frac{n}{t}}\Rightarrow$$$$f\left(t\right)=\frac{\pi }{2}{\int }_1^t\frac{e^{-\frac{n}{z}}}{z}dz+c....becontinued...$$

Answered by mind is power last updated on 23/Dec/19

$$\mathrm{f}\left(\mathrm{t}\right)={\int }_0^{+\mathrm{\infty }}\frac{{\tan }^{-}\left(\mathrm{xt}\right)\cos \left(\mathrm{nx}\right)}{\mathrm{x}}\mathrm{dx}$$$${\mathrm{f}}^{\prime }\left(\mathrm{t}\right)={\int }_0^{+\mathrm{\infty }}\frac{\cos \left(\mathrm{nx}\right)}{1+{\mathrm{x}}^2{\mathrm{t}}^2}\mathrm{dx}$$$$=\mathrm{n}{\int }_0^{+\mathrm{\infty }}\frac{\cos \left(\mathrm{z}\right)}{{\mathrm{n}}^2+{\mathrm{z}}^2{\mathrm{t}}^2}.\mathrm{dz}$$$$=\frac{\mathrm{n}}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{\cos \left(\mathrm{z}\right)}{{\mathrm{n}}^2+{\mathrm{z}}^2{\mathrm{t}}^2}\mathrm{dz}=\mathrm{Re}\left(\frac{\mathrm{n}}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{iz}}}{{\mathrm{n}}^2+{\mathrm{z}}^2{\mathrm{t}}^2}\mathrm{dz}\right)\mathrm{by}\mathrm{residu}\mathrm{th}$$$$\mathrm{f}\left(\mathrm{z}\right)=\frac{{\mathrm{e}}^{\mathrm{iz}}}{{\mathrm{n}}^2+{\mathrm{z}}^2{\mathrm{t}}^2}$$$$=\frac{\mathrm{n}}{2}.2\mathrm{i}\pi \left\{\mathrm{Res}(\mathrm{f},\frac{\mathrm{ni}}{\mathrm{t}})\right)\}$$$$\mathrm{in}\pi \frac{{\mathrm{e}}^{-\frac{\mathrm{n}}{\mathrm{t}}}}{2\mathrm{int}}\}=\frac{\pi }{2\mathrm{t}}{\mathrm{e}}^{-\frac{\mathrm{n}}{\mathrm{t}}}$$$$\mathrm{f}\left(0\right)={\int }_0^{+\mathrm{\infty }}\frac{\mathrm{arvtan}\left(0\right)\cos \left(\mathrm{nx}\right)\mathrm{dx}}{\mathrm{x}}=0$$$$$$$$\mathrm{f}\left(1\right)=\mathrm{\Delta }={\int }_0^1{\mathrm{f}}^{\prime }\left(\mathrm{t}\right)\mathrm{dt}={\int }_0^1\frac{\pi }{2\mathrm{t}}\left({\mathrm{e}}^{\frac{-\mathrm{n}}{\mathrm{t}}}\right)\mathrm{dt}=\frac{\pi }{2}{\int }_0^1\frac{{\mathrm{e}}^{-\frac{\mathrm{n}}{\mathrm{t}}}}{\mathrm{t}}\mathrm{dt}$$$$\mathrm{w}=\frac{\mathrm{n}}{\mathrm{t}}\Rightarrow \mathrm{dt}=-\mathrm{n}\frac{\mathrm{dw}}{{\mathrm{w}}^2}$$$$\mathrm{\Delta }=-\frac{\mathrm{n}\pi }{2}{\int }_1^{+\mathrm{\infty }}\frac{{\mathrm{e}}^{-\mathrm{w}}}{\mathrm{w}}\mathrm{dw}$$$$\mathrm{Ei}\left(\mathrm{x}\right)=-{\int }_{-\mathrm{x}}^{+\mathrm{\infty }}\frac{{\mathrm{e}}^{-\mathrm{t}}}{\mathrm{t}}\mathrm{dt},\mathrm{x}<0$$$$\mathrm{\Delta }=-\frac{\mathrm{n}\pi }{2}{\mathrm{E}}_{\mathrm{i}}(-1)$$$$$$$$$$

Commented by Ajao yinka last updated on 31/Dec/19

Perfect

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