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 recen...

Students of elementary complex analysis usually begin by seeing the derivation of the Cauchy–Riemann equations. A topic of interest to both the development of the theory and its applications is the reconstruction of a holomorphic function from its real part, or the extraction of the imaginary part from the real part, or vice versa. Usually this takes place by solving the partial differential sy...

1. The notation is standard. N, Z, R, R +, C denote respectively the natural, integral, real, real positive, and complex numbers. Re ~ and Im ~ is the real and the imaginary part of ~EC". [1 is the usual norm in C". By g(t), tER + u {+ oo}, we denote the C = functions defined for Ix[<t, xER", and by Co(t ) the C ~ functions with support in IxI<t. g(t) is endowed with the usual topology. If I" i...