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# hw1(1) - answer should depend on the value of s 4 Property...

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Homework assignment 1 * Due date: Monday February 12, 2007 1. Laguerre’s equation The Laguerre equation is: xy ′′ + (1 - x ) y + ny = 0 , where n is a non-negative integer. Show that for every n , Laguerre’s equation has a polyno- mial solution of degree n , and determine these polynomials for n = 0 , 1 , 2 and 3. 2. Method of Frobenius I Find the first 3 terms of the series expansion about x = 0 of 2 linearly independent solutions to x 2 y ′′ - x 2 y + ( x 2 - 2) y = 0 3. Method of Frobenius II Determine the form of a series expansion about x = 0 of 2 linearly independent solutions to xy ′′ - sy + x 3 y = 0 , where s is an arbitrary real number. Don’t determine the coefficients of the series. Your
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Unformatted text preview: answer should depend on the value of s . 4. Property of the Gaussian hypergeometric function. Denoting the Gaussian hypergeometric function by F ( α, β, γ ; x ), show that ln(1 + x ) = xF (1 , 1 , 2;-x ) . 5. Properties of Bessel functions. Denoting the Bessel function of the ±rst kind of order ν > 0 by J ν ( x ), show that the following properties hold: d dx ( x − ν J ν ( x ) ) =-x − ν J ν +1 ( x ) and J ν +1 ( x ) = 2 ν x J ν ( x )-J ν − 1 ( x ) . * MAP 4305; Instructor: Patrick De Leenheer. 1...
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• Summer '06
• DeLeenheer
• linearly independent solutions, Bessel function, Special functions, Hypergeometric series, Gaussian hypergeometric function

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