On Physics of asymptotic series and their (re)summation

There was a question on Physics Overflow which can be reduced to the last phrase:

I think the bottom line with my questions is that I fully accept that divergent series occur in physics all the time, and quite clearly they contain information that we can extract, but I would like to understand more to what degree can we trust those results.

And there were some answers including mine. Amongst other things, I mentioned a “constructive way” of building asymptotic series, which would be useful in practice. As an example, I considered a toy function E(x)=\int_0 ^\infty\frac{e^{-t}}{1+x\cdot t}dt\approx 1-x+2!x^2-3!x^3. A direct summations of its Taylor series is useless because it diverges from the exact function value at any finite x. It is so not only for fast “growing” coefficient cases, but also for regular (converging) Taylor series truncated at some finite order, when we try to extrapolate the truncated series to finite (large) values of x. A truncated series “grows” as the highest power of x, but the expanded function can be finite and limited, so the truncated series becomes inaccurate.

Thinking this fact over, in about 1981-1982, I decided that the difficulties with extrapolation to finite x was in expanding a finite and slowly changing function in powers of fast growing functions like x^n. “Why not to expand such a slow function in powers of slowly changing functions?“, thought I, for example: E(x)\approx 1-f(x)+a\cdot f(x)^2-b\cdot f(x)^3, where f(x)\to x at small x, but f(x)\ll x for finite x.

In order to give some idea and demonstrate fruitfulness of my “constructive” approach, I considered the following functions instead of x: Y(x)=\ln(1+x), Z(x)=x/(1+x), and f(x)=x/(1+k\cdot x) with adjustable coefficient k. The corresponding figures are the following:


Fig. 1. Expansion in powers of x.


Fig. 2. Expansion in powers of Y(x)=\ln(1+x).


Fig. 3. Expansion in powers of Z(x)=x/(1+x).

The smaller terms in the new series, the better approximation. Following this banal observation, I adjusted the coefficient k in f(x) to minimize the coefficient b at f^3 (note, the axis x is made longer in the next three figures):


Fig. 4. Expansion in powers of f(x)=x/(1+2x).

We see that E(x) is approximated now much better than with its truncated asymptotic series in powers of x (Fig. 1).

The same idea applied to the function I(g)=\int_{-\infty}^{\infty}e^{-x^2-gx^4}dx works well too:


Fig. 5. Expansion in powers of f(x)=0.75x/(1+4.375x).

Finally, the ground state energy E(\lambda) of the anharmonic oscillator in QM (anharmonicity \lambda x^4) has also a divergent series: E(\lambda)\approx 0.5(1+1.5\lambda -5.25\lambda ^2+41.625\lambda^3), which can be transformed into a series in powers of f(\lambda)=1.5\lambda/(1+3.5\lambda). It gives a good extrapolation of E(\lambda) (error \le \pm 1.5% within 0\le \lambda \le 1, Fig. 6), unlike the original series:


Fig. 6. The ground state energy of 1D anharmonic oscillator.

Thus, my idea was not too stupid as it allowed to extrapolate the asymptotic (divergent) series in the region of big x with a decent accuracy.

P.S. In my practice I also encountered a series f(\xi)\approx f(0)+f'(0)\xi+f''(0)\xi^2/2+... (a convergent one) whose convergence I managed to improve with partially summing up some of its terms into a finite function g(\xi), so the resulting series f(\xi)\approx g(\xi)+a\xi +b\xi^2+... became even better convergent (Chapters 3 and 4). It is somewhat similar to soft contributions summation in QED, if you like.

The moral is the following: if you want to have a “convergent” series, then build it yourself and enjoy.


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