= Hyperbolic cosine
{title2=$\cosh$}
The <hyperbolic cosine> is the <even function>
$$
\cosh x=\frac{e^x+e^{-x}}2.
$$
It has <derivative> $\sinh x$ and second <derivative> $\cosh x>0$, so it is a strictly <convex function>. Its unique <global minimum> is $\cosh0=1$, and its <image> is $[1,\infty)$. For example, $\cosh(-1)=\cosh1\approx1.543081$. On the unit <hyperbola>, the <hyperbolic cosine> is the horizontal coordinate; as a graph against $x$, it is a symmetric bowl.
Its <Taylor series> begins
$$
\cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots.
$$
The positive fourth-order term explains why the bowl resembles a <parabola> near its minimum yet differs from one away from it. Restricting to $x\geq0$ gives the <inverse hyperbolic cosine>.
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