Integral of orthogonal Bessel functions

Question

This is the basic form of the problem i am solving.

The main difference with the integral on the photo and my example is that i have my integral defined from b to c instead of 0 to a, and Bessel function definition is written in a simpler way.

FullSimplify[
  Integrate[
   r BesselJ[0, Subscript[s, n] r] BesselJ[0, Subscript[s, m] r], {r, 
    b, c}]] // TraditionalForm

The code is really basic, but i can't define that sn and sm are orthogonal in mathematica, even though they are in reality.

How does this change in integral borders and Bessel function definition affect integral result?

P.S.: This is the equation used to numerically define values of ss (or s).

Answer 1

The OP's integral is missing information about the subscripted variables (which I avoid) and the change of coordinates leading to the interval from b to c, which information is necessary to evaluate the integral. Here's the integral from the quote:

res = Integrate[
  r BesselJ[0, BesselJZero[0, n] r/a] BesselJ[0, 
    BesselJZero[0, m] r/a], {r, 0, a}, 
  Assumptions -> {m, n} ∈ PositiveIntegers]

FullSimplify yields a generically true result:

FullSimplify[res, {m, n} ∈ PositiveIntegers]
(*  0  *)

The exception is clear from the denominator; one can also use FullSimplify[FunctionSingularities[res, {m, n}], {m, n} ∈ PositiveIntegers] programmatically in theory, but there does not seem to be enough information about BesselJ encoded in Mathematica to reach the conclusion m == n.

Limit[res, m -> n, Assumptions -> n ∈ Integers] // FullSimplify

Answer 2

This is really a mathematics question rather than a Mathematica question, but from Abramowitz and Stegun section 11.4 we have

$\int_a^b t C\left[\nu ,\lambda _m t\right] C\left[\nu ,\lambda _n t\right] \, dt=0$ for $m\neq n$

and for m = n, a>b, the indefinite integral is

$\frac{1}{2} t^2 \left\{\left(1-\frac{\nu ^2}{\lambda _n^2 t^2}\right) C\left[\nu ,\lambda _n t\right]{}^2+C'\left(\nu ,\lambda _n t\right){}^2\right\}$

Apply the limits on $t$ at $a$ and $b$ to get the definite integral.

In the above $C[\nu ,z]=A J_{\nu }(z)+B Y_{\nu }(z)$

And $\lambda _n$ is a real root of $h_1 \lambda C[\nu +1,\lambda b]-h_2 C[\nu ,\lambda b]=0$

and you must have k's that fit $k_1 \lambda _n C\left[\nu +1,\lambda _n a\right]-k_2 C\left[\nu ,\lambda _n a\right]=0$.

Since you don't tell us exactly what problem you are trying to solve, it is hard to know whether your problem can be make to fit these parameters. You mention your solution is a combination of J0 and Y0 bessel functions, but are only integrating the J's, which makes no sense to me. It almost looks like you are trying to solve a bounded circular reservoir problem.