Holomorphic Polar Coordinates and Segal-Bargmann Space

In Real Euclidean Space, polar coordinates allow mathematicians to calculate the norm of higher dimensional SO-invariant functions with relative ease by reducing the problem to a 1-dimensional integral. In this dissertation I look at the Complex Segal-Bargmann Space using the C_t transform. I find there is a “holomorphic” version of polar coordinates that allows us to do the same in the odd dimensional cases. A geometric approach for this was done by Areerak Kaewthep and Wicharn Lewkeeratiyutkul using the B_t transform in [9], but this method is not easily generalized to non-Euclidean Spaces. Motived by the works of Gestur Olafsson and Henrik Schlichtkrull in [10], I use shift operators to find this “holomorphic” version of polar coordinates in C_t version of the Segal-Bargmann transform.