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L15 - Regular Singular Points Consider the SOLODE P x y 00...

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Unformatted text preview: Regular Singular Points Consider the SOLODE P ( x ) y 00 + Q ( x ) y + R ( x ) y = 0 ( * ) where P,Q,R are analytic about x and so many common factors x- x have been removed that at least one of P,Q,R does not vanish at x . Definition: A point x is an Ordinary point for ( * ) if P ( x ) 6 = 0 . Definition: A point x is a Singular Point for our SOLODE P ( x ) y 00 + Q ( x ) y + R ( x ) y = 0 ( * ) if P ( x ) = 0 . x is a Regular Singular Point for ( * ) if P ( x ) = 0 and in if addition α def = lim x → x ( x- x ) Q ( x ) P ( x ) and β def = lim x → x ( x- x ) 2 R ( x ) P ( x ) exist. ( RS ) Regular singular points occur for example in the quantum physics of the hydrogen atom and heavier atoms. 2 3 4 For α 6 = 0 6 = β α def = lim x → x ( x- x ) Q ( x ) P ( x ) and β def = lim x → x ( x- x ) 2 R ( x ) P ( x ) exist. ( RS ) means that ( x- x ) Q ( x ) P ( x ) and ( x- x ) 2 R ( x ) P ( x ) are power series whose constant terms are α and β , respectively: ( x- x ) Q ( x ) P ( x ) = α 1+ α 1 ( x- x )+ α 2 ( x- x ) 2 + ... ( x- x ) 2 R ( x ) P ( x ) = β 1+ β 1 ( x- x )+ β 2 ( x- x ) 2 + ... 5 This turns our SOLODE P ( x ) y 00 + Q ( x ) y + R ( x ) y = 0 for x 6 = x into y 00 + Q ( x ) P ( x ) | {z } p ( x ) y + R ( x ) P ( x ) | {z } q ( x ) y = 0 ⇐⇒ ( x- x ) 2 y 00 +( x- x ) h ( x- x ) Q ( x ) P ( x ) i y + h ( x- x ) 2 R ( x ) P ( x ) i y =0 ⇐⇒ ( x- x ) 2 y 00 +( x- x ) α [1+ α 1 ( x- x...
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L15 - Regular Singular Points Consider the SOLODE P x y 00...

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