  Holomorphic functions that characterize states in a two-dimensional Landau
level been central to key developments such as the Laughlin state. Their origin
has historically been attributed to a special property of "Schr\"odinger
wavefunctions" of states in the "lowest Landau level". It is shown here that
they instead arise in any Landau level as a generic mathematical property of
the Heisenberg description of the non-commutative geometry of guiding centers.
When quasiperiodic boundary conditions are applied to compactify the system on
a torus, a new formula for the overlap between holomorphic states, in the form
of a discrete sum rather than an integral, is obtained. The new formula is
unexpected from the previous "lowest-Landau level Schr\"odinger wavefunction"
interpretation.
