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# 5.4we - series about x is equivalent to the boundedness of...

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5.4: REGULAR SINGULAR POINTS KIAM HEONG KWA Recall that a point x 0 is said to be a singular point of the equation (1) P ( x ) d 2 y dx 2 + Q ( x ) dy dx + R ( x ) y = 0 , where P ( x ), Q ( x ), and R ( x ) are analytic functions about x 0 , provided one of the functions (2) p ( x ) = Q ( x ) P ( x ) and q ( x ) = R ( x ) P ( x ) is not analytic at x 0 . In the case P ( x ), Q ( x ), and R ( x ) are polynomials, this is equivalent to saying that P ( x 0 ) = 0. A singular point x 0 of (1) is said to be regular if and only if both ( x - x 0 ) p ( x ) = ( x - x 0 ) Q ( x ) P ( x ) , (3a) ( x - x 0 ) 2 q ( x ) = ( x - x 0 ) 2 R ( x ) P ( x ) (3b) have convergent Taylor series about x 0 . Otherwise, x 0 is said to be irregular . By treating the involved functions as functions of a complex variable, the condition that both the functions in (3) have convergent Taylor

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Unformatted text preview: series about x is equivalent to the boundedness of the functions in a deleted neighborhood of x , i.e., a subset of the xy-plane of the form { ( x,y ) ∈ R 2 | < p ( x-x ) 2 + y 2 < δ } for some δ > 0. In the case P ( x ), Q ( x ), and R ( x ) are polynomials, this is equivalent to requiring that the limits lim x → x ( x-x ) p ( x ) = ( x-x ) Q ( x ) P ( x ) , (4a) lim x → x ( x-x ) 2 q ( x ) = ( x-x ) 2 R ( x ) P ( x ) (4b) exist and are ﬁnite. Date : February 8, 2011. 1...
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5.4we - series about x is equivalent to the boundedness of...

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