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Q. 66

Found in: Page 681


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

The second-order differential equation


where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by Jp(x). It may be shown that Jp(x)is given by the following power series in x :


What is the interval of convergence for J1(x)?

The series is converges for all values of x.

See the step by step solution

Step by Step Solution

Step 1. Given information

An expression is given as Jp(x)=k=0(-1)kk!(k+p)!22k+px2k+p

Step 2. Interval of convergence

The J1(x) is


We have to do ratio test first for the absolute convergence,

Assume that bk=(-1)kk!(k+1)!22k+1x2k+1


It implies that

role="math" localid="1649407841088" limkbk+1bk=limk(-1)k+1(k+1)!(k+2)!22k+3x2k+3(-1)kk!(k+1)!22k+1x2k+1=limk-1(k+1)(k+2)4x2

Calculate for k tending to infinity,


Limit is zero independently of x. So the series is converges for all values of x.

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