Applied Bessel functions. by Relton, Frederick Ernest

By Relton, Frederick Ernest

Show description

Read Online or Download Applied Bessel functions. PDF

Similar mathematics_1 books

Mathematics, Affect and Learning: Middle School Students' Beliefs and Attitudes About Mathematics Education

This e-book examines the ideals, attitudes, values and feelings of scholars in Years five to eight (aged 10 to fourteen years) approximately arithmetic and arithmetic schooling. essentially, this publication specializes in the improvement of affective perspectives and responses in the direction of arithmetic and arithmetic studying. moreover, it appears scholars enhance their extra adverse perspectives of arithmetic throughout the center institution years (Years five to 8), and so the following we be aware of scholars during this severe interval.

Extra info for Applied Bessel functions.

Sample text

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.

Download PDF sample

Rated 4.76 of 5 – based on 7 votes