Chebyshev polynomials boost the π-series convergence

초록

We present an interesting application of the solution to the simple harmonic oscillator (SHO) that can serve as a computation of pi . We begin with a review of a compact teaching strategy for solving its equation of motion through integration in a general physics course, where many students face difficulties with conventional methods for solving differential equations. This integration approach leads to the arcsine function, the inverse of the sine function, ultimately providing the solution to the SHO. We investigate various series for approximating pi, focusing on the arcsine series and their difference in convergence speed. We begin with Newton's arcsine series for pi = 2arcsin1 . We then explore a series based on powers of sin pi/2(k+1)<< 1 , where k is a large positive integer and the sine term is computed using nested radicals through half-angle formulas, resembling Viete's formula. The small sine term acts as a power-counting parameter, making the series better convergent to pi with reliable error estimation. We extend this approach to a fractional-angle method, generalizing the factor from 1/2 to 1/p' for a prime number p', by employing Chebyshev polynomials of the second kind, which commonly arise in physics problems. This leads to a series involving powers of sin pi/p , where p is an arbitrary integer expressed as a product of prime factors, further enhancing convergence with a smaller power-counting parameter. The power counting allows us to identify significant terms in the Chebyshev polynomials and to truncate numerically insignificant contributions that optimize and simplify the computation of the sine term. Our novel strategies are pedagogical and suitable for advanced physics undergraduates, enabling them to approximate pi with high accuracy using techniques covered in physics courses.

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