Hyperbolic Cosine Function is Even
Theorem
Let $\cosh: \C \to \C$ be the hyperbolic cosine function on the set of complex numbers.
Then $\cosh$ is even:
- $\map \cosh {-x} = \cosh x$
Proof 1
| \(\ds \map \cosh {-x}\) | \(=\) | \(\ds \frac {e^{-x} + e^{-\paren {-x} } } 2\) | Definition of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{-x} + e^x} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^x + e^{-x} } 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh x\) |
$\blacksquare$
Proof 2
| \(\ds \map \cosh {-x}\) | \(=\) | \(\ds \map \cos {-i x}\) | Hyperbolic Cosine in terms of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {i x}\) | Cosine Function is Even | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cosh x\) | Hyperbolic Cosine in terms of Cosine |
$\blacksquare$
Also see
- Hyperbolic Sine Function is Odd
- Hyperbolic Tangent Function is Odd
- Hyperbolic Cotangent Function is Odd
- Hyperbolic Secant Function is Even
- Hyperbolic Cosecant Function is Odd
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.15$: Functions of Negative Arguments
- 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
- 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