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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 wellbehaved 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 onedimensional space of real analytic solutions to (1.17), which
are wellbehaved as x → 0.
• This onedimensional 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.
 Winter '14
 Equations

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