I'm computing this integral
$$\lim_{\varepsilon \to 0}\int_{|x|>\varepsilon}\frac{e^{-(t-x)^2}}{x}dx=?$$
I'm not sure that its integral whether exist. How could I solve it?
Thanks for attention!
2026-05-15 17:25:11.1778865911
$\lim_{\varepsilon \to 0}\int_{|x|>\varepsilon}\frac{e^{-(t-x)^2}}{x}dx=?$
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This is the Hilbert transform $$PV\int_{-\infty}^{\infty}\frac{e^{-x^2}}{t-x}dx=-i\pi e^{-t^2}Erf(it).$$ See M.L. Glasser, JCAM [\bf 10}, 293 (1984).