  The simplicity and the efficiency of a quasi-analytical method for solving
nonlinear ordinary differential equations (ODE), is illustrated on the study of
anharmonic oscillators (AO) with a potential $V(x) =\beta x^{2}+x^{2m}$
($m>0$). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE
with two-point boundaries (one being located at infinity), the solution of
which has singularities in the complex plane of the independent variable $x$. A
conformal mapping of a suitably chosen angular sector of the complex plane of
$x$ upon the unit disc centered at the origin makes convergent the transformed
Taylor series of the generic solution so that the boundary condition at
infinity can be easily imposed. In principle, this constraint, when applied on
the logarithmic-derivative of the wave function, determines the eigenvalues to
an arbitrary level of accuracy. In practice, for $\beta \geq 0$ or slightly
negative, the accuracy of the results obtained is astonishingly large with
regards to the modest computing power used. It is explained why the efficiency
of the method decreases as $\beta $ is more and more negative. Various aspects
of the method and comparisons with some seemingly similar methods, based also
on expressing the solution as a Taylor series, are shortly reviewed, presented
and discussed.
