A question for inverting the Hilbert transform

[Jiaqi-knight]

Hello sir,
Your class is worth learning in detail. Thank you for your kindness!!
In lecture 19, when you derive the solution of logarithmic singular integral equations, in the end, you say "recall, we can express the solution as".
What confused me is how to reach it? Recall which equation?
Is any interval give this equation? My only reachable reference is your paper "Computing the Hilbert transform and its inverse". Here, when in U unit circle, H^-1=-H.

Yours,
Jiaqi

[dlfivefifty]

That's the inverse Hilbert formula

[Jiaqi-knight]

Thank you, I was a little hard understand of the weighted term in the equation.

[dlfivefifty]

You mean the diamond? That's just the dummy variable

[Jiaqi-knight]

I mean why the inverse Hilbert is not just like this, 𝑢=H[f']+C, because of -H=H^-1

[dlfivefifty]

That’s the real line Hilbert transfoem