Need help solving equation involving $\cosh$ I am trying to solve this equation for $a$
$$R= (a)\cosh\left(\frac{l}{a}\right)$$
where $R$ and $l$ are real positive constants.
I tried breaking $\cosh$ into exponentials but this didnt seem to help.
 A: You might look at the Lambert W function  It is basically the only tool we have for things that combine polynomials and exponentials.  Whether you consider a solution in terms of that acceptable is a matter of taste.  Many people don't accept it.  I didn't look whether I could solve this in terms of W.   We could define a function $M(R)$ that gives the answer you want, but that wouldn't help much.  A numerical solution is probably what you are in for.
A: Note that there is a bit of scaling that can be done to remove one parameter: if $t = a/\ell$ and $r = R/\ell$, the equation becomes $r  = t \cosh(1/t)$.
Now, besides numerical methods, you might try a series solution (for one of the two  branches of the solution): for large $r$,
$$ t = r- \dfrac{1}{2! r}-\dfrac{7}{4! r^3}-\dfrac{241}{6! r^5}-\dfrac{19279}{8! r^7} + \ldots $$
A: $\large\mbox{Hints}:$


*

*
$\displaystyle{%
\mbox{Define}\ \mu \equiv {R \over \ell}\,,
\quad
x \equiv {\ell\ {\rm sgn}\left(\mu\right) \over a}.\
\mbox{Then,}\quad \left\vert\mu\right\vert\, x = \cosh\left(x\right) 
}$.


*
$\displaystyle{%
\mbox{The solutions satisfy}\ x > {1 \over \left\vert\mu\right\vert}  
}$.


*
There is $\it one$ solution $\left(~x_{t}~\right)$ which is valid for a particular valor $\mu_{t}$ of $\mu$. $x_{t}$ and $\mu_{t}$ are determined by the pair of equations
$$
\left.%
\begin{array}{rcl}
\cosh\left(x_{t}\right) & = & \left\vert\mu_{t}\right\vert\, x_{t}
\\[1mm]
\sinh\left(x_{t}\right) & = & \left\vert\mu_{t}\right\vert
\end{array}\right\}
\quad\mbox{which yield}\quad
\left\{%
\begin{array}{rcl}
\left\vert\mu_{t}\right\vert
& = &
\sinh\left(\sqrt{1 + {1 \over \mu_{t}^{2}}\,}\,\right)
\\[1mm]
x_{t} & = & \sqrt{1 + {1 \over \mu_{t}^{2}}\,}
\end{array}\right.
$$


*When $\left\vert\mu\right\vert < \left\vert\mu_{t}\right\vert$, there is not any real solution.

*
When $\left\vert\mu\right\vert > \left\vert\mu_{t}\right\vert$, we have $\it two$ solutions $\left(~x_{< \atop >}~\right)$:
$$
{1 \over \left\vert\mu\right\vert} <\ x_{<}\ <\ x_{t}\,,
\qquad\qquad
x_{>}\ >\ x_{t}
$$


*
When $\left\vert\mu\right\vert \gg 1,\quad$
$\displaystyle{%
x_{<} \approx {1 \over \left\vert\mu\right\vert}
+
{1 \over \left\vert\mu\right\vert^{3}}
}$



At least, the reader can take from here toward a
$\it\mbox{numerical calculation}$.
