Definition:Hyperbolic Cosine

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

• Equation of Catenary

• 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