hw1(4) - Homework assignment 1 Due date: September 18 1....

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Unformatted text preview: Homework assignment 1 Due date: September 18 1. Legendre's equation Find the first 7 terms of the power series expansion about x = 0 of the general solution to Legendre's equation: (1 - x2 )y - 2xy + p(p + 1)y = 0, where p is an arbitrary real parameter. These general solutions are called Legendre's functions. Explain why they are called Legendre polynomials in case p is a positive integer. 2. Method of Frobenius I Determine the form of a series expansion about x = 0 of 2 linearly independent solutions to xy - sy + x3 y = 0, where s is an arbitrary real number. Your answer should depend on the value of s. 3. Method of Frobenius II Find the first 3 terms of the series expansion about x = 0 of 2 linearly independent solutions to x2 y - x2 y + (x2 - 2)y = 0 4. Property of the Gaussian hypergeometric function. Denoting the Gaussian hypergeometric function by F (, , ; x), show that ln(1 + x) = xF (1, 1, 2; -x). 5. Properties of Bessel functions. Denoting the Bessel function of the first kind of order > 0 by J (x), show that the following properties hold: 2 d x- J (x) = -x- J+1 (x) and J+1 (x) = J (x) - J-1 (x). dx x MAP 4305; Instructor: Patrick De Leenheer. 1 ...
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