Hyperbolic Functions Notes

Overview of hyperbolic functions

• Hyperbolic functions arise as the hyperbolic counterparts to the circular trigonometric functions. The primary ones are sinh, cosh, and tanh, built from exponentials.
• Definitions (via exponentials):
  
  \sinh x = \frac{e^{x} - e^{-x}}{2}, \quad \cosh x = \frac{e^{x} + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}.
  
• Key qualitative idea: sinh and cosh are not periodic and sinh is unbounded in both directions, unlike sine which is bounded and periodic.
• They are intimately linked to the exponential function, but working with the sinh/cosh forms is often clearer for calculus (derivatives and identities).