Translations:Dictionary:Hilbert transform/76/en

The appearance of the function $-i\operatorname {sgn}(\omega )$ may seem a bit mysterious. Using this function is the only way to combine the result from both integrations into a unified expression. Each integration defines a result that is only good in the respective half-plane where the respective integral exists. The function $i\operatorname {sgn}(\omega )$ is the analytic continuation of these two integrations. This function is valid for all $\omega$ except possibly $\omega =0.$