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Unformatted text preview: nctions.3 Mathematica represents the Bessel functions of
the ﬁrst kind symbolically by BesselJ[n,x]. Thus to plot the Bessel function
Jn (x) on the interval [0, 15] one simply types in
n=0; Plot[ BesselJ[n,x], {x,0,15}]
and a plot similar to that of Figure 1.1 will be produced. Similarly, we can
plot Jn (x), for n = 1, 2, 3 . . . . Note that the graph of J0 (x) suggests that it has
inﬁnitely many positive zeros.
On the open interval 0 < x < ∞, Bessel’s equation has a twodimensional
space of solutions. However, it turns out that when p is a nonnegative integer, a
second solution, linearly independent from the Bessel function of the ﬁrst kind,
3 For a very brief introduction to Mathematica, the reader can refer to Appendix A. 25 0.6 0.4 0.2 2 4 6 8 10 12 14 0.2 Figure 1.2: Graph of the Bessel function J1 (x).
cannot be obtained directly by the generalized power series method that we have
presented. To obtain a basis for the space of solutions, we can, however, apply
the meth...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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