Derivation of a New Analytical Formula for the Second Order Bessel Function of the First Kind

In this study, a new analytical approximation for the Bessel function of the first kind of order two J2(t), t>0, is produced using its power series and its asymptotic series as t tends to infinity. The absolute error between the new analytical approximation and the function J2(t) is analyzed for t∈[0,104], and the results indicate that it decreases significantly as t increases for large values of t. Additionally, our new approximation formula presents estimations for the positive zeros of the function J2(t) with extremely minor relative errors; the largest relative error for the first positive zero is 3.83594×10−5 and, thereafter, the relative errors gradually decrease until they reach 9.91627×10−11 for the twelfth zero. We also showed that our findings outperform some recently published ones.