The eigenvalue equation is $\lambda^2 + \lambda a e^{-\lambda \tau_c} + b e^{-\lambda \tau_c} = 0$ where $\lambda$ is a variable, and other parameters including $a \in R$, $b \in R$ and $\tau_c \in R$ are constants. I want to get the closed form of the eigenvalue roots for the eigenvalue equation by using the Lambert W Function Approach. Then, these eigenvalue roots are used for the stability analysis of multi-dimensional time-delay systems. But I don't know how to solve it.
Any help will be appreciated.
1)
$$\lambda^2+\lambda ae^{-\lambda\tau_c}+be^{-\lambda\tau_c}=0$$
Because your equation contains two algebraically independent monomials ($\lambda,e^{-\lambda\tau_c}$), there is possibly no non-constant (partial) inverse that is an elementary function.
But if the equation can have solutions that are elementary numbers is a different mathematical problem.
If $\tau_c=0$, we have
$$\lambda^2+a\lambda+b=0$$
with the solutions
$$-\frac{1}{2}a+\frac{1}{2}\sqrt{a^2-4b},-\frac{1}{2}a-\frac{1}{2}\sqrt{a^2-4b}.$$
If $(a,b,\tau_c\text{ algebraic })\land (\tau_c\neq 0)$, the equation doesn't have solutions that are elementary numbers.
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2)
For applying Lambert W, we want to try to rearrange your equation to the form
$$F(f(\lambda)e^{f(\lambda)})=F(y),$$
where $F$ and $f$ are unary functions in the complex numbers, and $y$ is independent of $\lambda$.
$$\lambda^2+\lambda ae^{-\lambda\tau_c}+be^{-\lambda\tau_c}=0$$
$$(\lambda a+b)e^{-\lambda\tau_c}=-\lambda^2$$
$$\frac{(\lambda a+b)e^{-\lambda\tau_c}}{\lambda^2}=-1$$
We see, the general equation isn't in a form that allows to apply Lambert W. But it can be solved by Generalized Lambert W:
$$\lambda=-\frac{1}{\tau_c}W\left(^{\frac{b}{a}\tau_c}_{0,\ 0};\frac{1}{a\tau_c}\right)=\frac{1}{\tau_c}W\left(^{0,\ \ 0}_{-\frac{b}{a}\tau_c};a\tau_c\right)$$
$-$ see the references below.
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For $a=0$, we get the solutions
$$\frac{2}{\tau_c}W_{k_1}(\frac{1}{2}\sqrt{-b\tau_c^2}),\frac{2}{\tau_c}W_{k_2}(-\frac{1}{2}\sqrt{-b\tau_c^2})\ \ \ \ \ \ \ \ \ \ (k_1,k_2\in\mathbb{Z}).$$
For $b=0$, we get the solutions
$$0,\frac{1}{\tau_c}W_{k_3}(-\frac{a}{\tau_c})\ \ \ \ \ \ \ \ \ \ (k_3\in\mathbb{Z}).$$ $\ $
[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. 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 generalized Lambert W functions. 2018
[Stoutemyer 2022] Stoutemyer, D. R.: Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses. 2022