# A biholomorphism from the Bell representative domain onto an annulus and kernel functions

#### Abstract

Let *A*_{ρ2,1} = {* z* ∈ C : ρ^{2} < :*z*: < 1} and let Ω* _{ r}* = {

*z*∈ C : :

*z*+ 1z : < r}. It is known that for

*r*> 2, Ω

*is a doubly-connected domain with an algebraic Bergman kernel and satisfies a certain quadrature identity. We find an explicit biholomorphism between these two domains, and get the value of ρ as a function of*

_{ r}*r,*which follows from the computation of the biholomorphism. Using the transformation formula for the Bergman kernel, we write out the Bergman kernel of the Bell representative domain Ω

*, which was earlier known to be algebraic. We also determine the Ahlfors maps of this generalized quadrature domain using the Ahlfors map of the annulus. ^*

_{r}#### Degree

Ph.D.

#### Advisors

Steven R. Bell, Purdue University.

#### Subject Area

Mathematics

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