Asymptotic decomposition and asymptotic formula for the root of the transcendental equation with a parameter

We consider the classic finite equation containing a parameter. Under a certain condition on the left side of this equation after replacing the variable it is reduced to the kind that it is not difficult to classify the interrelations between its constituent parts. Every case entails a certain situation with the existence of the solution of the equation under study, and it is shown that it can have, in essence, the same standard form. For the latter one fundamental result of the construction of the asymptotic decomposition is given. Next, the proof of formula for coefficients of the desired decomposition is presented using inductive technique. Another approach to finding a solution of the specified equation is associated with the possibility of obtaining an asymptotic formula in appearance resembling an infinite continued fraction. At first, approximations are naturally constructed recursively as consistently refined inequalities for the solution, and then, the convergence of these approximations is strictly proved. The pointwise convergence of separately even and odd approximations is related to their monotony and limitations, and the additional condition of continuous differentiability of the equation’s incoming data also guarantees uniform convergence of approximations to the solution. In conclusion, a simple example of such continued fraction is given.