Numerical calculation of bessel functions by solving differential equations and its application

The method solving Bessel's differential equation for calculating numerical values of the Bessel function J(v)(x) is not usually used, but it: is made clear here that the differential equation method can give very precise num erical values of J(v)(x), and is very effective if we do not mind computing time, Here we improved the differential equation method by using a new tra nsformation of J(v)(x). This letter also shows a method of evaluating the e rrors of J(v)(x) calculated by this method. The recurrence method is used f or calculating the Bessel function J(v)(x) numerically. The convergence of the solutions in this method, however, is not yet examined for all of the v alues of the complex v and the reals. By using the differential equation me thod, this letter will numerically ascertain the convergence of the solutio ns and the precision of J(v)(x) calculated by the recurrence method.