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Question Number 37356 by math khazana by abdo last updated on 12/Jun/18

$$letb\in CandRe\left(b\right)>0provethat$$ $${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{iax}}{x-ib}dx=2i\pi e^{-ab}and$$ $${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{iax}}{x+ib}dx=0$$

Commented bymath khazana by abdo last updated on 12/Jun/18

$$a>0$$

Commented bymath khazana by abdo last updated on 13/Jun/18

$$let\phi \left(z\right)=\frac{e^{iaz}}{z-ib}soibisasimplepoleof\phi$$ $$withRe\left(b\right)>0\Rightarrow Im\left(ib\right)>0{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi \left(z\right)dz=2i\pi Res(\phi ,ib)$$ $$=2i\pi e^{ia\left(ib\right)}=2i\pi e^{-ab}$$ $$let\psi \left(z\right)=\frac{e^{iaz}}{x+ib}thepoleof\psi is-ibbutIm(-ib)<0$$ $$so{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\psi \left(z\right)dz=0$$

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