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sh

Represents the hyperbolic sine function, commonly used in mathematical analysis and engineering calculations.

Overview

Essential in advanced mathematics and physics for describing exponential growth and oscillatory behavior, particularly in differential equations and complex analysis.

  • Frequently appears in solutions to wave equations and fluid dynamics
  • Used in electrical engineering for signal analysis
  • Important in the study of hyperbolic functions alongside cosh and tanh
  • Common in problems involving sinh-cosh identities and hyperbolic transformations

Examples

Hyperbolic sine function in a basic trigonometric equation.

sh⁡(x)=ex−e−x2\sh(x) = \frac{e^x - e^{-x}}{2} 
\sh(x) = \frac{e^x - e^{-x}}{2}

Relationship between hyperbolic sine and regular sine.

sh⁡(ix)=isin⁡(x)\sh(ix) = i\sin(x) 
\sh(ix) = i\sin(x)

Identity involving hyperbolic sine in a mathematical expression.

ddxsh⁡(x)=ch⁡(x)\frac{d}{dx}\sh(x) = \ch(x) 
\frac{d}{dx}\sh(x) = \ch(x)