Mathematical Properties, Stability Theory, and Experimental Validation of the Double Exponential Activation Function in Complex-Valued Neural Networks

Complex-valued neural networks (CVNNs) can naturally process complex signals, phase information, and two-dimensional data, and have been widely applied in various fields. However, Liouville's theorem states that a bounded entire function must be constant, which implies that there exists no nontrivial activation function that is both bounded and holomorphic everywhere. Consequently, the design of activation functions for CVNNs must trade off between holomorphicity and boundedness. We introduce a family of double exponential functions and focus on the activation function σ(z)=e-eᶻ. We establish the complex-analytic foundations, proving that σ is an entire function, uniformly bounded in the strip |Im(z)|<π/2 (0<|σ(z)|≤1), with a bounded gradient and tightness. Furthermore, we reveal its Taylor series, real symmetry, periodicity, translation-scaling law, and connections with modified Bessel functions, Chebyshev polynomials, and the Gumbel distribution. On this basis, we establish a formal definition of complex-valued neural networks based on σ, an imaginary-part clipping operator, holomorphic and continuous function approximation theorems, gradient flow analysis, a convergence rate of $O(1/K)$, linear convergence under the Polyak-Łojasiewicz (PL) condition, and generalization bounds. Experiments on MNIST show that the double-e network achieves higher test accuracy than CReLU and modReLU, with lower test loss; ablation studies confirm that imaginary-part clipping is a necessary condition for numerical stability; parameter sweeps find that β is the most critical parameter—the smaller β, the wider the bounded region and the higher the accuracy; and noise robustness experiments reveal a fundamental trade-off between phase preservation and noise robustness in the design of complex activation functions. This provides CVNNs with an activation function that combines holomorphicity and strip boundedness, and establishes a corresponding theoretical framework with experimental validation.