Wallis formula from the harmonic oscillator

Generalizing Wallis' Formula

The present note generalizes Wallis’ formula, 2  =  . 7 6 . 5 6 . 5 4 . 3 4 . 3 2 . 1 2 , using the EulerMascheroni constant g and the Glaisher-Kinkelin constant A: 2 ln 2 4 = 3 3 2 . 1 2 . 5 4 3 4

Padé approximant related to the Wallis formula

Based on the Padé approximation method, in this paper we determine the coefficients [Formula: see text] and [Formula: see text] such that [Formula: see text] where [Formula: see text] is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results.

Harmonic Oscillator

Radiation from a uniformly accelerating harmonic oscillator

We consider a radiation from a uniformly accelerating harmonic oscillator whose minimal coupling to the scalar field changes suddenly. The exact time evolutions of the quantum operators are given in terms of a classical solution of a forced harmonic oscillator. After the jumping of the coupling constant there occurs a fast absorption of energy into the oscillator, and then a slow emission follo...

Special value formula for the spectral zeta function of the non-commutative harmonic oscillator

This series is absolutely convergent in the region Rs > 1, and defines a holomorphic function in s there. We call this function ζQ(s) the spectral zeta function for the non-commutative harmonic oscillator Q, which is introduced and studied by Ichinose and Wakayama [1]. The zeta function ζQ(s) is analytically continued to the whole complex plane as a single-valued meromorphic function which is h...