## Applied Bessel functions. by Relton, Frederick Ernest By Relton, Frederick Ernest

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So that nJn + xJn' = xJn_v (3) corresponding to the recurrence formula 3-1(3) in the last chapter. The whole o f the results proved for cylinder functions can therefore be applied to Jn(x). ^x), 2r {xnJn(x)} = •7-i = «70 = {x~nJn{x)} — —x-nJn+l(x), x n- rJ n -r{x ), *71> 48 APPLIED BESSEL FUNCTIONS and so on. 5 Since Bessel’s equation is unaltered if —n replaces n, we conclude that J -n(x) is equally a solution. The factor outside the bracket is enough to show that it is not a mere numerical multiple of Jn(x); it is accordingly an independent solution and the general solution is y = AJn(x) + BJ_n(x).

Hence 10 jx ? J 22d x = cc6( J 22 + J 32), 10jx_5J 32iZx = —x~\J£ -f- J 32). APPLIED BESSEL FUNCTIONS 54 EXERCISES 1. Prove that the indicia! equation for x^y" - f xy' + (x2 — n2)y = 0 is r2 = n* and deduce the series for Jn(x), 2. Verify that the second root of the indicial equation in the text, r = — 2n, leads to the series for J^n(x). 3. Verify that the expansion for Jn(x) is convergent for all values of x. Prove that it is absolutely convergent. dx 4. Prove that the solution of Bessel's equation is AJn -f- BJnJ 5.

Touch the #-axis ? In what circumstances could a solution 1 1 sin2# 4. Prove that the last equation has the normal form v" — v ------------- = 0. 1 — sin2# 5. Prove that the equation of damped oscillations, # + 2a# - f (a2 + b2)x = 0 has the same normal form as when the oscillations are undamped. y' / 1\ 6. Reduce the equation y" + — + (1 — ^ J y = 0 to normal form. What do you conclude from the result ? 7. Prove that any solution of x2y" + xy' + (#2 — n2)y = 0 must have an infinity of zeros. 8.