Question about the Frobenius method and Bessel functions

[timetraveller123]

Homework Statement

i have been trying to learn bessel function for some time now but to not much help

firstly, i don't even understand why frobenius method works why does adding a factor of x^r help to fix the singularity problem. i saw answers on google like as not all function can be represented like negative,fractional,complex exponents and adding the r helps to take of .

secondly, why does frobenius method only work when it is regular singular point. from what i saw
$y'' + \frac{p(x)}{x}y' + \frac{q(x)}{x^2}y = 0$
p and q need to analytic around zero why this condition. why wouldn't frobenius work if they weren't

thirdly, from what i see in the previous equation, for such equations the second derivative is not defined at zero. but when i see bessel function(solution of one such equation) the second derivative seems to be well defined at x =0

how is it not defined .

i am sorry if i don't make sense but i am very much a begineer in these thing it would be of much help if anyone could help me out thanks

Homework Equations

The Attempt at a Solution

 

[bolbteppa]

If the coefficient functions in your ode have pole singularities, then one would expect the solution may also have pole singularities, which is why you'd throw a $x^r$ out in front of a power series so that your solution can at most be a finite Laurent series. Have a look at section 10.3 of

https://archive.org/details/courseofmodernan00whit/page/197

to see why Frobenius will not give two solutions without the conditions you mentioned (the indicial equation is then at most of first order).

 

[timetraveller123]

If the coefficient functions in your ode have pole singularities, then one would expect the solution may also have pole singularities

wait what are you saying solutions to such equations also have "problems" at that point because once again as an example the bessel function seems to behave nicely at x=0
i am really sorry if i don't make sense and i really appreciate your help thanks
and i will look through the book you said

 

[pasmith]

wait what are you saying solutions to such equations also have "problems" at that point because once again as an example the bessel function seems to behave nicely at x=0
i am really sorry if i don't make sense and i really appreciate your help thanks
and i will look through the book you said

As with every second-order linear ODE, Bessel's equation has two linearly independent solutions. The J function, which is the one you have been looking at, is analytic at the origin. The other solution is the Y function, which has a singularity at the origin.

For this reason it is often ignored in applications.

 

[timetraveller123]

oh wow thanks that clears up a lot i completely forgot about the second kind
but why does r have to be constant for every term what if the exponents in the power series of the solutions are not separated by integers
maybe what i don't understand is how does one ensure that the new r term will encompass all such functions