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I am looking for a source that provides as many examples as possible of distributions induced by principal value integrals similar to 1, 2, and [3]. Any suggestions? If not, please let me know of any examples you know with the final value and I'll work it out

Morcus
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    Those three are closely related: $\operatorname{pv}\frac{1}{x} = (\ln|x|)'$ and $\operatorname{fp}\frac{1}{x^2} = - (\operatorname{pv}\frac{1}{x})'.$ You can of course continue taking derivatives. – md2perpe Nov 20 '21 at 16:14
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    If $f\in C^\infty$ has a zero of degree 1 at $a$ then $\frac{1}{f}$ is a distribution of pv-type in a neighborhood of $a$. One such example is $f=\sin.$ – md2perpe Nov 20 '21 at 16:28
  • Thanks. You're right. But these examples are overused. I'm looking for something that I haven't seen yet. – Morcus Nov 20 '21 at 22:58
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    Perhaps a one-sided like the derivative of $H(x) \ln(x)$? Also, there are some examples of finite part distributions in the Lecture Notes by Hasse Carlsson. – md2perpe Nov 20 '21 at 23:12
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    I think that $\frac{1}{x}\ln|x|$ can be interpreted as a distribution of pv-type. – md2perpe Nov 21 '21 at 14:51

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