M257-316Notes_Lecture7

M257-316Notes_Lecture7 - Chapter 6 Lecture 7 Frobenius...

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Chapter 6 Lecture 7 - Frobenius Series about Regular Singular Points Example 1: Ly =2 x 2 y 0 xy 0 +(1+ x ) y =0 x = 0 is a RSP. y = X n =0 a n x n + r (6.1) Ly x 2 X n =0 a n ( n + r )( n + r 1) x n + r 2 x X n =0 a n ( n + r ) x n + r 1 x ) X n =0 a n x n + r X n =0 a n { 2( n + r )( n + r 1) ( n + r )+1 } x n + r + X n =0 a n x n + r +1 (6.2) m = n +1 n m =1 n = m 1 Therefore a 0 { 2 r ( r 1) r } x r + X n =1 [ a n { 2( n + r )( n + r 1) ( n + r } + a n 1 ] x n + r . 37
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Lecture 7 - Frobenius Series about Regular Singular Points x r > Indicial Equation: 2 r 2 3 r +1=(2 r 1)( r 1) = 0 r = 1 2 ,r =1. a 0 arbitrary Recursion a n = a n 1 (2 n +2 r 3)( n + r )+1 (6.3) Let r =1 / 2: a n = a n 1 (2 n 2)( n +1 / 2)+1 = a n 1 ( n 1)(2 n +1)+1 = a n 1 n (2 n 1) n =1: a 1 = a 0 1 ; n =2: a 2 = a 1 2 . 3 = + a 0 2 . 3 a 3 = a 2 3 . 5 = a 0 1 . (2 . 3)(3 . 5) ; a 4 = a 3 4 . 7 = + a 0 1(2 . 3)(3 . 5)(4 . 7) (6.4) a n = ( 1) n n !1 . 3 . 5 . (2 n 1) y 1 ( x )= x 1 / 2 X n =0 ( 1) n n ! x n 1 . 3 . 5 . (2 n 1)= x 1 / 2 X n =0 ( 1) n 2 ( n 1) n !
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M257-316Notes_Lecture7 - Chapter 6 Lecture 7 Frobenius...

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