Hyperbolic Tangent Function is Odd
Theorem
Let $\tanh: \C \to \C$ be the hyperbolic tangent function on the set of complex numbers.
Then $\tanh$ is odd:
- $\map \tanh {-x} = -\tanh x$
Proof
| \(\ds \map \tanh {-x}\) | \(=\) | \(\ds \frac {\map \sinh {-x} } {\map \cosh {-x} }\) | Definition 2 of Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\sinh x} {\map \cosh {-x} }\) | Hyperbolic Sine Function is Odd | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\sinh x} {\cosh x}\) | Hyperbolic Cosine Function is Even | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\tanh x\) | Definition 2 of Hyperbolic Tangent |
$\blacksquare$
Also see
- Hyperbolic Sine Function is Odd
- Hyperbolic Cosine Function is Even
- 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): $8.16$: 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: $5$