THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

634 Appendix B. Bessel FunctionsB.4 Approximations for Small Argument x < 1J 0 → 1 − x 24J 1 → x 2 − x 3Y 0 → 2 π ln xY 1 →− 2 πB.5 Approximations for Large Arguments x > 2πH (1)nH (2)nB.6 Recursion Relations1x16√2J n →(xπ x cos − nπ 2 − π )4√2Y n →(xπ x sin − nπ 2 − π )4√2→π x ei(x− nπ 2 − π 4 )√2→π x e−i(x− nπ 2 − π 4 )The relations listed below hold true for C being any of the Bessel functions of thefirst, second, or third kind or for linear combinations of these functions.C n+1 + C n−1 = 2nx C nddxddxdC 0dx =−C 1dC ndx = 1 2 (C n−1 − C n+1 )(x n )C n = x n C n−1( ) 1=− 1 x n x C n n+1