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Question Number 30321 by rahul 19 last updated on 20/Feb/18

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{a}x}}{{\mathrm{e}}^x+1}dx=?$$

Commented by rahul 19 last updated on 20/Feb/18

$${\mathrm{what}}^{\prime }\mathrm{s}\mathrm{residus}\mathrm{theorem}?$$

Commented by abdo imad last updated on 20/Feb/18

$$ch.e^x=t\iff x=lnt\Rightarrow I={\int }_0^{\mathrm{\infty }}\frac{t^a}{t+1}\frac{dt}{t}={\int }_0^{\mathrm{\infty }}\frac{t^{a-1}}{1+t}dt$$$$thisintegralconverges\iff 0<a<1andweproveby$$$$residustheoremthatI=\frac{\pi }{sin\left(\pi a\right)}.$$

Commented by abdo imad last updated on 20/Feb/18

$$forthistheoremtherearemanycaseslettakeforexample$$$$F\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}withpandqpolynoials/{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p\left(x\right)}{q\left(x\right)}dxis$$$$convergentandq\left(x\right)haverootsnotreals{\left(z_i\right)}_{1⩽i⩽p}so$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p\left(x\right)}{q\left(x\right)}dx=2i\pi \sum _{z\in P^{+}}Res(F,z)with$$$$P^{+}=\{z\in C/zpoleofFandim\left(z\right)>0\}butyoumusttake$$$$alookinthebookofcomplexanalysis...$$

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