Vessel equation Questions

Without beating through your working (which seems over-complex) we need to say that there is a solution to the Bessel equation such that C zero is non zero.

Once we have that solution written as a polynomial each coefficient in the polynomial must be separately zero.
i.e. if the solution is ax + bx^2 + cx^3 ... then a=0, b = 0, c=0 etc and each of a, b, c will be a polynomial in r.

So work out the bits (I write C rather than C sub zero to save having to latex this)

y=Cx^r, y'=Crx^(r-1) and y''=Cr(r-1)x^(r-2)

when these are substituted into Bessel's equn we get for terms involving C zero (again just using C). We are only interested in the C zero term

y(x) = Cx^r [ r (r-1) + r + (x^2 - m^2)] + other terms not involving C zero

Now multiplying this out (and rejecting the term in x^(r+2) cos that is not an x^r term) we get the coefficient of x^r is C[r^2 - r + r -m^2]

If C is non-zero then for the Solution to be valid the r term in the coefficient of x^r must be zero.

i.e. [r^2 - r + r -m^2] must be zero which gives us r^2=m^2

Hope that's clear

Well this falls out after some algebra.
Realise that the coefficients if x^(r+n) mainly involve Cn but the x^2-m^2 gives an -m^2 Cn x^(r+n) but the x^2 picks out C(n-2) x^(r+n).

$Cn [(x^2-m^2)+(r+n) + (r+n)(r+n-1)]x^{r+n}+...$


When we multiply out the x^2 we end up with a term in x^(r+n+2) and so we have to pick out a term in Cn-2 to find the x^(r+n) term.

Sorry if that is not clear. Having a terrible time with getting tex to cooperate with subscripts

Thank you I needed this tip will try and solve further later today.

You might also find this helpful:

http://mathworld.wolfram.com/FrobeniusMethod.html

The example they use on the method is the Bessel equation - lucky for you - but don't just copy it out or you won't learn anything...