Systematic perturbation calculation of integrals with applications to physics

In this paper we generalize and improve a method for calculating the period of a classical oscillator and other integrals of physical interest, which was recently developed by some of the authors. We derive analytical expressions that prove to be more accurate than those commonly found in the literature, and test the convergence of the series produced by the approach.

Figure 1

Logarithmic plot of the error $\mid b_0^{\left(N\right)}-b_0\mid ∕b_0$ as a function of the order $N$. Diamonds and crosses correspond to present results and those of Pelster et al., respectively. The dashed and dotted lines correspond to the linear fits of each data. The circles correspond to the results of [7].

Figure 2

Logarithmic plot of the error $\mid {\omega }_2-{\omega }_{\mathit{\text{exact}}}\mid ∕{\omega }_{\mathit{\text{exact}}}$ as a function of the order $\mu A^2$. The three curves correspond to the second order results for the present method, the LPLDE method of [5, 7], and the method of Pelster et al., respectively. The horizontal lines are the asymptotic values of the errors (see Fig. 1).

Figure 3

Logarithmic plot of the error $\mid b_0^{\left(N\right)}-b_0\mid ∕b_0$ as a function of the order $N$ for the quadratic sextic oscillator.

Figure 4

Precession of the orbit of a planet (in seconds of degree) calculated assuming $M=1.97\times {10}^{30}\mathrm{kg}$, $G∕c^2=7.425\times {10}^{-30}\mathrm{m}∕\mathrm{kg}$, and $ϵ=0.2506$ as a function of $a$ (i.e., the average of perihelion and aphelion) for values close to a black hole. The solid line is the exact result; the thin line corresponds to the asymptotic formula; the other lines correspond to the different orders to which our approximation has been applied.

Figure 5

Logarithmic plot of the error over the precession of the perihelion as a function $a$. The different lines correspond to the different orders to which our approximation has been applied.