Hyperbolic Sine Function is Odd
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Contents
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
\(\displaystyle \map \sinh {-x}\) | \(=\) | \(\displaystyle \frac {e^{-x} - e^{-\paren {-x} } } 2\) | Definition of Hyperbolic Sine | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {e^{-x} - e^x} 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle -\frac {e^x - e^{-x} } 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle -\sinh x\) |
$\blacksquare$
Proof 2
\(\displaystyle \map \sinh {-x}\) | \(=\) | \(\displaystyle -i \, \map \sin {-i x}\) | Hyperbolic Sine in terms of Sine | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle i \, \map \sin {i x}\) | Sine Function is Odd | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle -\sinh x\) | 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: $5$