The Bergman kernel function and the Szego-kernel function

We compute the holomorphic derivative of the harmonic measure associated to a C-infinity bounded domain in the plane and show that the exact Bergman kernel function associated to a C-infinity bounded domain in the plane relates the derivatives of the Ahlfors map and the Szego kernel in an explicit way. We find several formulas for the exact Bergman kernel and the Szego kernel and the harmonic measure. Finally we survey some other properties of the holomorphic derivative of the harmonic measure.