I have worked for quite a while on a statistical problem and has been able to simplify the problem to an equation with a variable I have been seeking, $q$.
The equation is $$ e^{-q} = \alpha q $$
where $q$ and $\alpha$ are
- $ \alpha = e^{-(\theta_d + \theta_{\rho}\frac{r_{n}t}{n} + \theta_i + 1)}$
- $ q = 1 + \theta_d(1-\hat{d}) $
and, $\hat{d} \in (0,1]$ is the actual value I am interested to have on closed form. Everything is real valued, but the specifics doesnt really matter so wont go into detail.
From what i've gathered this is called a trancendental function and typically requires numerical solutions. My question is if there is anything I can do with this equation? Are there any tricks to solve this for $q$ or $\hat{d}$ ? Is there anything I can infer from $\alpha$ for when this equation does and does not have a supported solution?
$$e^{-q}=aq$$
The equation is an algebraic equation (a polynomial equation) of more than one algebraically independent monomials ($q,e^{-q}$). We cannot rearrange the equation for $q$ by elementary functions we can read from the equation therefore.
Lambert W is one of only a few Special functions that can invert polynomial equations with two algebraically independent monomials. Lambert W is not an elementary function.
For applying Lambert W, we have to try to rearrange the equation to $f(q)e^{g(q)}=c$, where $c$ is a constant.
$$e^{-q}=aq$$ $$1=aqe^q$$ $$aqe^q=1$$ $$qe^q=\frac{1}{a}$$ $$q=W\left(\frac{1}{a}\right)$$