How to Integrate e^x / (1+x^2) with Complex Contour Integration?

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SUMMARY

The discussion focuses on integrating the function e^x / (1+x^2) using complex contour integration techniques. The integral presented is ∫_{-∞}^{∞} e^{iωt} / (1 + (t^2/τ^2)) dt, where i represents the imaginary unit, and ω and τ are positive real numbers. Participants emphasize the importance of the residue theorem from complex variables as a critical tool for solving this integral effectively.

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Einsling
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Hi
Could you please tell me how to integrate it? Thanks ~~

[tex]\int_{-\infty}^{\infty} \frac {e^{i\omega t}} {1+\frac{t^2}{\tau^2}} dt[/tex]

where i is imaginary unit, [tex]\omega, \tau[/tex] are positive real,
 
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Complex contour integration should be good. Are you familiar with the residue theorem from complex variables?
 

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