Sum of Hyperbolic Sine and Cosine equals Exponential
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Theorem
Let $z \in \C$ be a complex number.
Then:
- $e^z = \cosh z + \sinh z$
where:
- $e^z$ denotes the complex exponential function
- $\cosh z$ denotes the cosine function
- $\sinh z$ denotes sine function
Proof
\(\ds \cosh z + \sinh z\) | \(=\) | \(\ds \dfrac {e^z + e^{-z} } 2 + \dfrac {e^z - e^{-z} } 2\) | Definition of Hyperbolic Cosine and Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {e^z + e^z + e^{-z} - e^{-z} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^z\) |
$\blacksquare$
Also see
Sources
- 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