monotonous series: In the smoothing function f$_{p}$ ( x ) in Eq.(10) the most important part of the smoothing is produced by the central region in h$_{p}$ ( x ): x ∈ [ − x ( p ) 1 , x ( p ) 1 ], determined by the first root x ( p ) 1 . One can see in the figures that for p > 0 values x ( p +2) 1 < x ( p ) 1 i.e. the value of x ( p ) 1 decreases when p increases. The finite range smoothing has the advantage that the effect of a certain single particle energy e$_{i}$ vanishes beyond the interval E ∈ [ e$_{i}$ − γ, e$_{i}$ + γ ]. Therefore, the smoothed level density becomes exactly zero for energies lying below ( e$_{1}$ − γ ), while the Gaussian oscillates around zero. This oscillation character appears at any value of the smoothing parameter. If we go to higher E -values, we can smooth the oscillatory character of the ˜ g ( E ) if we use large enough γ values in the smoothing function with Gaussian weight function. This is not the case however, if we smooth with finite range weight function, where some undulation in ˜ g ( E ) remains even if we use large smoothing range parameters. Therefore, it can not be well approximated by a straight line as it was in the generalized Strutinski procedure. This seems to be an important difference between the smoothed level densities calculated by using Gaussian or finite range smoothings. We calculate the smoothed energy in Eq. (13) by using the finite range smoothing functions, for a range of γ ∈ [ γ$_{min}$, γ$_{max}$ ] and p ∈ { p$_{min}$, p$_{min}$ + 2 , ..., p$_{max}$ } values. This allows us to study the plateau curves. For p = 0 the plateau curve is an monotonously increasing function, therefore, neither the plateau condition in Eq.(19) nor the local plateau condition can be applied. (There is no γ value where the derivative is zero.) This result show the necessity of using curvature correction polynomials. For p > 0 plateau curves have minima (and maxima) where the plateau condition in Eq.(19) is fulfilled locally. However the plateau curves might have several minima and we have to find the proper one among those minima. A necessary condition of the smoothing is that the smoothed level density should not reflect the shell structure of the single particle levels. Therefore, in the smoothing procedure we have to start searching for the minimum of δE ( γ, p ) from a ( p -dependent) γ$_{min}$ value with which the shell structure has already disappeared. The most important characteristics of the single particle spectrum is the largest gap between the occupied levels. Therefore, we have to determine the largest distance between 12 0 < x ( p ) 1 < x ( p ) 2 < ... < x ( p ) m . (30)