We first express Bessel's Equation in(adsbygoogle = window.adsbygoogle || []).push({}); Sturm-Liouville formthrough a substitution:

Next, we consider aseries solutionand replace v by m where m is an integer. We obtain a recurrence relation:

Then, since all these terms must be = 0,

Consider m = 0

First term vanishes and second term =

a_{1}x = 0

therefore, a_{1}= 0. Then by recurrence relation above, a_{1}= a_{3}= a_{5}= ..... = a_{2n-1}= 0

Then only the even series give non-zero coefficients, so we start with a_{0}≠ 0.

Consider m = 1

Second term vanishes, and first term =

a_{0}= 0

By recurrence relation, this implies a_{2n}= 0

So we start the series with a_{1}≠ 0

Book's Explanation

I don't understand why a_{0}= a_{1}= .... = a_{m-1}= 0

Surely, for any value of m that is ≠ 1 we have:

[tex]a_{1}x = \frac{m^2}{1 - m^2} a_0 [/tex]

This doesn't imply anything above? Also, how did they get the resulting Bessel functions? It looks wildly different from mine.

a_{0}= a_{1}= .... = a_{m-1}= 0,

[tex] u = \sum_{n=m}^\infty a_{n} x^n = \frac {1}{2^n n!} x^n [/tex]

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# Bessel Equation and Bessel fuctions

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