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I came across this for the first time today.

[itex]\int_0^\infty e^{i\omega t}dt = \pi\delta(\omega)+iP(\frac{1}{\omega})[/itex]

Here [itex]P(\frac{1}{\omega})[/itex] is the so called principal value. I haven't seen this term normally so can I ask where we get it from?

Googling principal value showed me a very different thing. It said that the principal value of an integral of a function is to take a sum of integrals such that we skip over those values where the function is not well defined. But here, the complex exponential is always well defined so what exactly is the reason for the second term?

Thank you :)