solutions for quiz xxix

Hence we obtain x j x j x xj x 0 bessels dierential

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 x− 1 8!! 2 9 10 8!! x8 − 8 2 x+ 1 10!! 11 12 10!! + J x + J ′′ x = 1 2 0!! 2 − 1 4 2!! 2 x2 + 1 6 4!! 2 x4 − 6 1 8 6!! 2 x6 + 1 10 8!! 2 x8 − 1 12 10!! 2 x10 + ··· . Realize that x multipled to this exactly equals −J ′ x . Hence we obtain x J ′′ x + J′ x + xJ x =0 Bessel’s differential equation . J x is called the Bessel function of the first kind . n2 ∞ [II] (10pts) (1) The radius of convergence for 1− n=1 1 n xn is R = e. n2 Work : Set aℓ = =⇒ R = 1− 1 n . Then n2 an an+1 = = 1− 1 n 1− 1 n+1 , (n+1)2 . n2 lim n− ∞ → an an+1 = lim n− ∞ → In order to compute this limit, take the logarithm: 7 1− 1− 1 n 1 n+1 (n+1)2 . n2 1− ln R = lim ln n− ∞ → 1− = = = = = + 1 n lim n− ∞ → lim n− ∞ → lim n− ∞ → lim n− ∞ → lim n− ∞ → 2 n ln 2 n ln 2 n ln 2 n ln 2 n n2 + 2n + 1 1 n+1 1− 1− 1− 1− − 1 n (n+1)2 − 1 n − 1 n + 1 n + n+1 n+1 n+1 n+1 2 2 2 2 ln ln ln ln 1− 1 n+1 n n+1 1 n n+1 n 1+ 1 1 1 1 1 − − − − − ·...
View Full Document

This note was uploaded on 01/12/2014 for the course MATH 116 taught by Professor Scholle,minho during the Fall '08 term at Kansas.

Ask a homework question - tutors are online