Definition:Hyperbolic Sine

Definition

The hyperbolic sine function is defined on the complex numbers as:

$\sinh: \C \to \C$:

$\forall z \in \C: \sinh z := \dfrac {e^z - e^{-z} } 2$

Real Hyperbolic Sine

On the real numbers it is defined similarly.

The real hyperbolic sine function is defined on the real numbers as:

$\sinh: \R \to \R$:

$\forall x \in \R: \sinh x := \dfrac {e^x - e^{-x} } 2$

Also see

• Definition:Hyperbolic Cosine
• Definition:Hyperbolic Tangent
• Definition:Hyperbolic Cotangent
• Definition:Hyperbolic Secant
• Definition: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

• 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): 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 Sine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicSine.html