Definition:Hyperbolic Sine/Real
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Definition
The real hyperbolic sine function is defined on the real numbers as:
- $\sinh: \R \to \R$:
- $\forall x \in \R: \sinh x := \dfrac {e^x - e^{-x} } 2$
Definition by Hyperbola
Definition:Real Hyperbolic Sine/Definition by Hyperbola
Also see
- Definition:Real Hyperbolic Cosine
- Definition:Real Hyperbolic Tangent
- Definition:Real Hyperbolic Cotangent
- Definition:Real Hyperbolic Secant
- Definition:Real Hyperbolic Cosecant
- Results about the hyperbolic sine function can be found here.
Linguistic Note
The usual symbol sinh for hyperbolic sine is awkward to pronounce.
Some pedagogues say it as shine, and some as sinch.
Others prefer the mouthful which is hyperbolic sine.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.1$: Definition of Hyperbolic Functions
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Definition of Hyperbolic Functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperbolic function
- Weisstein, Eric W. "Hyperbolic Sine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicSine.html