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The simplicity and the efficiency of a quasi-analytical method for solvingnonlinear ordinary differential equations (ODE), is illustrated on the study ofanharmonic 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 ODEwith two-point boundaries (one being located at infinity), the solution ofwhich has singularities in the complex plane of the independent variable $x$. Aconformal mapping of a suitably chosen angular sector of the complex plane of$x$ upon the unit disc centered at the origin makes convergent the transformedTaylor series of the generic solution so that the boundary condition atinfinity can be easily imposed. In principle, this constraint, when applied onthe logarithmic-derivative of the wave function, determines the eigenvalues toan arbitrary level of accuracy. In practice, for $\beta \geq 0$ or slightlynegative, the accuracy of the results obtained is astonishingly large withregards to the modest computing power used. It is explained why the efficiencyof the method decreases as $\beta $ is more and more negative. Various aspectsof the method and comparisons with some seemingly similar methods, based alsoon expressing the solution as a Taylor series, are shortly reviewed, presentedand discussed.

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators