Oberseminar Funktionentheorie, The dark side of the unit disk VI: Complex spherical harmonics (Annika Moucha)

We introduce complex spherical harmonics which are a generalization of the classical spherical harmonics in ℝ³. The complex spherical harmonics are holomorphic functions on the complex two sphere {(z₁,z₂,z₃)∈ℂ̂³: z₁²+z₂²+z₃²=1}. Since the complex two sphere is biholomorphic to the set  Ω={(z,w)∈ℂ̂²:zw≠1}we can use the theory discussed in the previous talks of this lecture series to explore the properties of the complex spherical harmonics. Among other properties of spherical harmonics we will place particular emphasis on the zonal harmonic functions which fulfill some invariance property with respect to isometries of the sphere.