Harmonic number

Abstract: First observed visible harmonics of CO/sub 2/-laser-irradiated plane and microballoon targets are reported. For intensities >5 x 10/sup 14/ W/cm/sup 2/, the harmonic production efficiency is constant over several visible harmonics with a cutoff at a high harmonic number. Two-dimensional particle simulations performed where there is a highly steepened density profile show a flat spectrum for high harmonics, with a cutoff at the harmonic where the upper density shelf is underdense. Harmonics thus offer a means of measuring the upper density shelf and its dynamics.

Abstract: The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler’s constant γ and its alternating analog ln (4/π), and on the other hand the infinite products of the first author for e, of the second author for π, and of Ser for e γ We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch’s transcendent of Hadjicostas’s double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions The main tools are analytic continuations of Lerch’s function, including Hasse’s series We also use Ramanujan’s polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values

Abstract: The small‐signal theory of a harmonic gyrotron is presented for the case of axis‐encircling electron orbits. The beam current required for oscillation is found to be highly dependent on the electron energy. Gyrotron cavities operating on this principle at very high harmonic numbers (n≂10) and high frequency in weak magnetic fields are well matched to low‐current, moderate‐energy, rf‐accelerated electron beams (≂50 mA, ≂250 keV), resulting in compact submillimeter wave systems.

Abstract: The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the second author for Euler's constant $\gamma$ and its alternating analog $\ln(4/\pi),$ and on the other hand the infinite products of the first author for $e$, and of the second author for $\pi$ and $e^\gamma.$ We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch's function, including Hasse's series. We also use Ramanujan's polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values.