Generalisation of Lambert W function?

Question

I want to solve an equation of the form:

$\exp(C / x) - 1 = D / (x + a)$

This seems to be almost in a form where I can express solutions in terms of the Lambert W function but I can't seem to figure it out myself. Can anyone help?

Thanks.

Answer 1

For applying only Lambert W and elementary functions, your equation should be in the form

$$f_1(f_2(x)e^{f_2(x)})=c,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$

where $c$ constant and $f_1$ and $f_2$ are elementary functions with a suitable elementary partial inverse.

$$e^{C/x}-1=D/(x+a)\ \ \ (x\neq -a,0)$$ $x\to\frac{1}{t}\ \ \ (t\neq 0)$: $$e^{Ct}-1=\frac{D}{\frac{1}{t}+a}$$ $$e^{Ct}=\frac{D}{\frac{1}{t}+a}$$ $$e^{Ct}=\frac{1+at+Dt}{1+at}$$ $$\frac{1+at}{1+at+Dt}e^{Ct}=1$$

Unfortunately your equation cannot be brought into the form of equation (1).
We see, Lambert W cannot be applied. But the equation is solvable by generalized Lambert W:

$$t=\frac{1}{C}W\left(^{\ -\frac{C}{a}}_{-\frac{C}{D+a}};\frac{D+a}{a}\right)=-\frac{1}{C}W\left(^{\frac{C}{D+a}}_{\ \frac{C}{a}};\frac{a}{D+a}\right)$$

$-$ see the references below. $\ $

[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equa-tion. J. Math. Anal. Appl. 455 (2017) (1) 538-553

[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)

[Castle 2018] Castle, P.: Taylor series for generali-zed Lambert W functions. 2018