Bergman kernel

Summary

The Bergman kernel of a domain is the reproducing kernel of its Bergman space $A^2$ of square-integrable holomorphic functions. It is a function $K(z, w)$, holomorphic in $z$ and antiholomorphic in $w$, such that every $f \in A^2$ satisfies $f(w) = \int f(z)\overline{K(z, w)}\, dA(z)$. For the unit disc it is $K(z, w) = \frac{1}{\pi (1 - z\bar w)^2}$. It gives the orthogonal projection from $L^2$ onto $A^2$, it transforms simply under biholomorphic maps, and in several variables it defines the Bergman metric.