TeXipedia

cth

Represents the hyperbolic cotangent function, commonly used in mathematical analysis and physics calculations.

Overview

Essential in advanced mathematics and theoretical physics, particularly when working with hyperbolic functions and differential equations.

  • Frequently appears in complex analysis and mathematical physics problems
  • Used in calculations involving wave equations and fluid dynamics
  • Often encountered alongside other hyperbolic functions like sinh and cosh
  • Important in the study of special functions and series expansions

Examples

Using the hyperbolic cotangent function in a calculus identity.

ddxcth⁡(x)=−1sinh⁡2(x)\frac{d}{dx} \cth(x) = -\frac{1}{\sinh^2(x)} 
\frac{d}{dx} \cth(x) = -\frac{1}{\sinh^2(x)}

Expressing a hyperbolic relationship in physics equations.

cth⁡(ωt)=cosh⁡(ωt)sinh⁡(ωt)\cth(\omega t) = \frac{\cosh(\omega t)}{\sinh(\omega t)} 
\cth(\omega t) = \frac{\cosh(\omega t)}{\sinh(\omega t)}

Simplifying a hyperbolic expression.

cth⁡2(x)−1=1sinh⁡2(x)\cth^2(x) - 1 = \frac{1}{\sinh^2(x)} 
\cth^2(x) - 1 = \frac{1}{\sinh^2(x)}