We present a new proof of the Poisson integral formula for harmonic functions using the methods of representation theory. In doing so, we exhibit the irreducible subspaces and unitary structure of a representation of the group SU(1,1) of 2 x 2 complex generalized special unitary matrices. Our arguments illustrate a technique that can be used to prove similar reproducing formulas in higher dimensions and for other classes of functions. Our paper should be accessible to readers with minimal knowledge of complex analysis.

Author Bio

Ewain Gwynne is a Junior at Northwestern University majoring in Mathematics and Mathematical Methods in the Social Sciences, along with a minor in Statistics. He hopes to pursue a PhD in math following the completion of his undergraduate degree. When not doing math of his own, Ewain tutors math and statistics, both online and in person. In his spare time (or, sometimes, whilst tutoring online) he also enjoys reading and playing Starcraft. This paper is the product of an REU project at Indiana University, completed during the summer of 2011.