Hyperbolic Sine Function is Odd
Theorem
Let $\sinh: \C \to \C$ be the hyperbolic sine function on the set of complex numbers.
Then $\sinh$ is odd:
- $\map \sinh {-x} = -\sinh x$
Proof 1
| \(\ds \map \sinh {-x}\) | \(=\) | \(\ds \frac {e^{-x} - e^{-\paren {-x} } } 2\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{-x} - e^x} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac {e^x - e^{-x} } 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\sinh x\) |
$\blacksquare$
Proof 2
| \(\ds \map \sinh {-x}\) | \(=\) | \(\ds -i \, \map \sin {-i x}\) | Hyperbolic Sine in terms of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds i \, \map \sin {i x}\) | Sine Function is Odd | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\sinh x\) | Hyperbolic Sine in terms of Sine |
$\blacksquare$
Also see
- Hyperbolic Cosine Function is Even
- 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): $8.14$: 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: $5$
- 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