Hyperbolic Function

Every function $f(x)$ can be written uniquely as sum of an odd part and an even part.

$f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x) - f(-x)}{2}$ $\therefore e^{x}=\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}$ The even and odd terms are called as hyperbolic cosine and hyperbolic sine function of $x$ .

i.e.
$\cosh x=\frac{e^x+e^{-x}}{2}$ $\sinh x=\frac{e^x-e^{-x}}{2}$

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$