3320hw5sol

3320hw5sol - Math 3320 Problem Set 5 Solutions 1 For each...

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Unformatted text preview: Math 3320 Problem Set 5 Solutions 1 For each of the first six problems, compute the value of the given periodic or eventually periodic continued fraction by first drawing the associated infinite strip of triangles, then finding a linear fractional transformation T in LF( Z ) that gives the periodicity in the strip, then solving T(z) = z . 1. 1 upslope ր 2 + 1 upslope ր 5 The periodicity transformation is the translation sending ( 1 / , / 1 ) to ( 1 / 2 , 5 / 11 ) , with matrix parenleftBig 1 5 2 11 parenrightBig of determinant 1. The equation T(z) = z is z + 5 2 z + 11 = z , or z + 5 = 2 z 2 + 11 z which simplifies to 2 z 2 + 10 z − 5 = 0, with roots ( − 5 ± √ 35 )/ 2. The value of the continued fraction is the positive root ( − 5 + √ 35 )/ 2. 2. 1 upslope ր 2 + 1 upslope ր 1 + 1 upslope ր 1 The transformation is a glide-reflection with matrix parenleftBig 1 2 3 5 parenrightBig of determinant − 1. This gives the equation z + 2 = 3 z 2 + 5 z , so 3 z 2 + 4 z − 2 = 0 and the positive root is ( − 2 + √ 10 )/ 3....
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This note was uploaded on 11/09/2009 for the course MATH 332 at Cornell.

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3320hw5sol - Math 3320 Problem Set 5 Solutions 1 For each...

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