# Bessel integral divergence

## Main Question or Discussion Point

Hi,

I would like to confirm my intuition about a bessel integral from you guys.

The integral is: Integrate[ (1/r) * J[2,2*pi*phi*r] ] from 0 → ∞ with respect to r.

J[2,2*pi*phi*r] is a second order bessel. Integrals with 1/x from 0 to Inf are divergent. Sure enough, this one is gonna diverge so mathematica says. But is there anything I might be able to do to stop this integral from diverging?

Thanks in advance for any insight!

Integrate[ (1/r) * J[2,2*pi*phi*r] ] = -J[1,2*pi*phi*r]/(2*pi*phi*r)
limit (r -> 0) = -1/2
limit (r -> infinity) = 0
The integral is convergent = -1/2

Integrate[ (1/r) * J[2,2*pi*phi*r] ] = -J[1,2*pi*phi*r]/(2*pi*phi*r)
limit (r -> 0) = -1/2
limit (r -> infinity) = 0
The integral is convergent = -1/2
Ah, oops i'm sorry. I had meant a 0th order bessel, not 2nd order! It's J[0,2*pi*phi*r]. =(

In this case, the integral is divergent when the lower boundary for r is =0. That is a fact and you can do nothing against a fact.
If you want to obtain a finite value, you have to chose a boundary higher than 0.