The Principle Value of a Singular Integral

Cauchy s Integral Formula states that if F z) is analytic within and on a closed contour C, then 

These relations are saying that w(a ,0), y(x,0) are infinite Hilbert transforms of each other. This topic, mainly oriented towards finite Hilbert transforms, is discussed, in more detail, in Sect. A2.4. Note that if F z) were analytic in the upper half-plane, the signs in (A2.1.8) would be the other way. 

---