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sinh

Represents the hyperbolic sine function, a fundamental exponential relationship used in mathematics and physics.

Overview

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

  • Commonly appears in differential equations, especially those modeling physical phenomena like wave propagation and electric fields
  • Used extensively in complex analysis and signal processing
  • Often paired with other hyperbolic functions like cosh and tanh in mathematical expressions
  • Crucial in the study of special relativity and electromagnetic theory
  • Appears frequently in engineering calculations involving exponential growth and decay patterns

Examples

Hyperbolic sine function in a basic identity.

sinh⁡x=ex−e−x2\sinh x = \frac{e^x - e^{-x}}{2} 
\sinh x = \frac{e^x - e^{-x}}{2}

Hyperbolic sine in a differential equation.

ddxsinh⁡x=cosh⁡x\frac{d}{dx} \sinh x = \cosh x 
\frac{d}{dx} \sinh x = \cosh x

Hyperbolic sine in a physics equation for catenary curve.

y=asinh⁡(xa)y = a\sinh\left(\frac{x}{a}\right) 
y = a\sinh\left(\frac{x}{a}\right)