A distributional version of Cauchy’s integral formula with applications to the Hilbert transformation

Holomorphic functions f ∈ H(C I) f ∈ H ( C \ I ) are represented by an “integral" over the jump of the associated distribution along the branch cut I . The “integral" is the evaluation of the Cauchy kernel in the sense of the duality between the locally convex spaces B₁(R) B 1 ( R ) and D'L¹,-1(R) D L 1 , - 1 ′ ( R ) . New Hilbert transforms of distributions are given.