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A Bergman kernel is a function of a Complex Variable with the ``reproducing kernel'' property
defined for any Domain in which there exist Nonzero Analytic Functions of class 
with respect to the Lebesgue Measure 
.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation
of the Soviet ``Mathematical Encyclopaedia.'' Dordrecht, Netherlands: Reidel, pp. 356-357, 1988.