Definition:Hyperbolic Cosine
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Definition
The hyperbolic cosine function is defined on the complex numbers as:
- $\cosh: \C \to \C$:
- $\forall z \in \C: \cosh z := \dfrac {e^z + e^{-z} } 2$
Real Hyperbolic Cosine
On the real numbers it is defined similarly.
The real hyperbolic cosine function is defined on the real numbers as:
- $\cosh: \R \to \R$:
- $\forall x \in \R: \cosh x := \dfrac {e^x + e^{-x} } 2$
Also see
- Definition:Hyperbolic Sine
- Definition:Hyperbolic Tangent
- Definition:Hyperbolic Cotangent
- Definition:Hyperbolic Secant
- Definition:Hyperbolic Cosecant
- Results about the hyperbolic cosine function can be found here.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.21)$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosh or ch
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hyperbolic function
- 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
- Weisstein, Eric W. "Hyperbolic Cosine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosine.html