The Integral and Limit Exchange Theorem for Hyperbolic Complex Columns

This paper delves into hyperbolic numbers and their generalizations like bicomplex and hyperbolic complex numbers. By analyzing prior studies, it explores their mathematical properties, including basic concepts, function sequences, and integral commutativity.The key finding is the proof of the theorem on the interchange of integral and limit of hyperbolic variable function numbers, which is significant for theoretical research.These numbers and functions have wide ranging applications in physics. For example, they are used in visible frequency hyperbolic plasmon polaritons and hyperbolic metamaterial based photonic weyl nodal line semimetals, providing new ideas for physics research.