Modified Euler-Maclaurin Formula And Its Applications

Euler-Maclaurin summation formula is modified for one dimension,which is further generalized to two dimensions.With the help of the modified Euler-Maclaurin summation formula,the dissociation of heavy quarkonia in relativistic heavy-ion collisions is calculated.Part of the remainder in the original Euler-Maclaurin formula is extracted and taken into calculation in the modified formula by expanding periodic Bernoulli polynomials into Fourier series.Applying the modified formula for one dimension twice successively,the formula is generalized to two dimensions.The modified formulae can be as precise as required if m,n,and p are chosen properly.Applications of the modified formulae are discussed,such as the calculation of thermal properties of a quantum rotator and a square well if two dimensions,and the evaluation of the Riemann ? function.We describe the quark-gluon plasma by the Bjorken hydrodynamics,which is as the background of the heavy quarkonium suppression described by the rate equation.The results in the constrained space are compared with that in the infinite space.The main difference comes from the change in the spectrum of gluons.The longitudinal momentum becomes discrete in the constrained space,and the energy of the ground state is finite,which leads to reduction in the dissociation rate of heavy quarkonia.Quantitative calculation requires a summation over the longitudinal momentum of gluons,which is achieved approximately by the previous modified Euler-Maclaurin formula.It is found that even if both the quark-gluon plasma and the heavy quarkonia are formed early enough,the dissociation at the early stage is still negligible.An effective initial time is estimated,which is inversely proportional to the binding energy of the quarkonium at low transverse momentum and increases with transverse momentum.