Hyperbolic Sine Function is Odd

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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


Sources