# Does a closed form solution for the value $a$ exist?

I'm having trouble trying to solve this algebraically for the value a, ie a = some function, where a, g and v are independent variables. Is there a closed form solution for this?

$g$ = $ln(1+a)$ - $\frac{0.5v}{(1+a)^2}$

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No, this is a transcendental equation in $a$. Have a look at Lambert W-Function.

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Thanks for the reply, I really appreciate it. –  yug tham Feb 19 at 3:19

If you make a change of variable such that, for example, $x=\frac{1}{(a+1)^2}$, the equation write $$2g=-\log (x)-{v}{x}$$ which is probably the best candidate for a Lambert function based solution as already mentioned by Ryuky. The solution of this last equation is given by $$x=\frac{W\left(e^{-2 g} v\right)}{v}$$ where $W$ is the Lambert function. Then, from $x$, we have $a$ from the definition.

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