Knowra Hyperbolic functions Hyperbolic functions Hyperbolic functions are functions defined from exponentials that parallel trigonometric functions. Their central members are hyperbolic sine, cosine, and tangent, linked by identities associated with the unit hyperbola.
Exponential function : A function in which the variable appears in the exponent, commonly written as eˣ. The exponential definitions of hyperbolic sine and cosine produce the whole family.
Hyperbolic sine : The function sinh x = (eˣ − e⁻ˣ)/2. It supplies the odd coordinate in the standard hyperbolic parameterization.
Catenary : The curve traced by a perfectly flexible, uniform chain hanging under gravity. Its profile is expressed using hyperbolic cosine.
Circular functions : The sine, cosine, and tangent functions associated with circular motion and the unit circle. Their squared sine-cosine identity uses a plus sign, unlike the hyperbolic identity.
Unit hyperbola : The curve x² − y² = 1 in the Cartesian plane. Hyperbolic cosine and sine parameterize its right-hand branch.
Hyperbolic cosine : The function cosh x = (eˣ + e⁻ˣ)/2. It supplies the even coordinate and satisfies cosh²x − sinh²x = 1.
Special relativity : The theory of space, time, and motion for inertial observers, developed by Einstein. Rapidity parameterizes Lorentz boosts through hyperbolic functions.
Complex trigonometric functions : Sine, cosine, and related functions extended to complex arguments. Imaginary arguments convert between circular and hyperbolic functions.
Trigonometric functions : Functions that relate angles to ratios and coordinates on the unit circle. Their familiar identities and names provide the analogy behind the hyperbolic functions.
Hyperbolic tangent : The function tanh x = sinh x / cosh x. Its bounded range follows from the exponential ratio and its denominator never vanishes on the real line.
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