1 eigenvalues of symmetric matrices before proceeding

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Unformatted text preview: ch is unbounded as x → 0. It follows that two linearly independent solutions y1 (x) and y2 (x) to Bessel’s equation cannot both be well-behaved at x = 0. Let us summarize what we know about the space of solutions to Bessel’s equation in the case where p is an integer: xW (y1 , y2 )(x) = c, or W (y1 , y2 )(x) = • There is a one-dimensional space of real analytic solutions to (1.17), which are well-behaved as x → 0. • This one-dimensional space is generated by a function Jp (x) which is given by the explicit power series formula x Jp (x) = 2 p∞ (−1)m m=0 1 m!(p + m)! x 2 2m . Exercises: 1.4.1. Using the explicit power series formulae for J0 (x) and J1 (x) show that d J0 (x) = −J1 (x) dx and d (xJ1 (x)) = xJ0 (x). dx 1.4.2. The differential equation x2 d2 y dy − (x2 + p2 )y = 0 +x dx2 dx is sometimes called a modified Bessel equation . Find a generalized power series solution to this equation in the case where p is an integer. (Hint: The power series you obtain should be very similar to the power series for...
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This document was uploaded on 01/12/2014.

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