Complex Hyperbolic and Trigonometric Functions

This lecture introduces complex hyperbolic functions as generalizations of real cases, defining cosh(z) = (exp(z) + exp(-z)) and sinh(z) = (exp(z) - exp(-z)). It explores the properties of these functions, such as cosh(z) - sinh(z) = 1, and their relationships with exponential functions. The lecture also covers complex trigonometric functions, defining cos(z) = cosh(iz) and sin(z) = -i sinh(iz), with identities like cos^2(z) + sin^2(z) = 1. It delves into the periodicity of these functions and their connections to exponential functions, providing insights into their behavior in the complex plane.