Non-constant bounded holomorphic functions of hyperbolic numbers - Candidates for hyperbolic activation functions
in Y. Kuroe, T. Nitta (eds.), Proceedings of The First SICE Symposium on Computational Intelligence [Concentrating on Clifford Neural Computing], 30 Sep. 2011, KIT, Kyoto, Japan, catalogue no. 11PG0009, pp. 23 - 28, 2011.
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.
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