  The Liouville theorem states that bounded holomorphic complex functions are
necessarily constant. Holomorphic functions fulfill the socalled Cauchy-Riemann
(CR) conditions. The CR conditions mean that a complex $z$-derivative is
independent of the direction. Holomorphic functions are ideal for activation
functions of complex neural networks, but the Liouville theorem makes them
useless. Yet recently the use of hyperbolic numbers, lead to the construction
of hyperbolic number neural networks. We will describe the Cauchy-Riemann
conditions for hyperbolic numbers and show that there exists a new interesting
type of bounded holomorphic functions of hyperbolic numbers, which are not
constant. We give examples of such functions. They therefore substantially
expand the available candidates for holomorphic activation functions for
hyperbolic number neural networks.
  Keywords: Hyperbolic numbers, Liouville theorem, Cauchy-Riemann conditions,
bounded holomorphic functions
