TeXipedia

coth

Represents the hyperbolic cotangent function, commonly used in mathematical analysis and differential equations.

Overview

Essential in advanced mathematics and physics calculations, particularly when dealing with hyperbolic functions and their applications.

  • Frequently appears in complex analysis and differential equations
  • Important in electromagnetic theory and wave propagation
  • Used in solving problems involving heat transfer and fluid dynamics
  • Often encountered alongside other hyperbolic functions like sinh and cosh
  • Valuable in engineering calculations involving exponential decay and growth

Examples

Definition of hyperbolic cotangent in terms of exponentials.

coth⁡(x)=cosh⁡(x)sinh⁡(x)=ex+e−xex−e−x\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}} 
\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}

Derivative of hyperbolic cotangent.

ddxcoth⁡(x)=−csch2(x)\frac{d}{dx} \coth(x) = -\text{csch}^2(x) 
\frac{d}{dx} \coth(x) = -\text{csch}^2(x)

Series expansion of hyperbolic cotangent around x.

coth⁡(x)=1x+x3−x345+O(x5)\coth(x) = \frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + O(x^5) 
\coth(x) = \frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + O(x^5)