# Integral convergent or divergent?

How can I find out for what values of r (both lower and upper limits), is this integral convergent/divergent?

I = Integrate[(1/r^2)*BesselJ[0, (m/R0)*r]*D[BesselJ[0, (m/R0)*r], {r, 2}], {r, ?, R0}]


Here m is the root of BesselJ[1, m] == 0

• In[29]:= FunctionPoles[ii, r] Out[29]= {{0, 2}} This shows that the integration path cannot contain the origin. Sep 26 at 14:58

## 2 Answers

Try transformation of variable r->R0 \[Rho] (assuming R0>0)

int=Integrate[(1/\[Rho]^2)*BesselJ[0, m \[Rho]]*
D[BesselJ[0, m \[Rho]], {  \[Rho], 2}], {\[Rho],  \[Rho]0 ,  1},
Assumptions -> { 0 < \[Rho]0 < 1}]/R0

(*(1/(2 R0 \[Rho]0))m^2 (2 \[Rho]0 HypergeometricPFQ[{-(1/2)}, {1,1}, -m^2] -
2 HypergeometricPFQ[{-(1/2)}, {1,1}, -m^2 \[Rho]0^2] - \[Rho]0
HypergeometricPFQ[{-(1/2), 3/2}, {1/2, 2, 2}, -m^2] +
HypergeometricPFQ[{-(1/2), 3/2}, {1/2, 2, 2}, -m^2 \[Rho]0^2])*)


Substitution of result m from BesselJ[1, m] == 0:

mi = Values@NSolve[{BesselJ[1, m] == 0, 0 <= m < 20}, m] // Flatten

Plot[Table[int R0 , {m, mi}], {\[Rho]0, 0, 1}, Evaluated -> True,AxesLabel -> {"\[Rho]0", "int R0"}]


Integral seems to be convergent for the values mi

• Does this mean that the integral is convergent in 0 < rho < infinity ? Sep 26 at 15:21
• No, convergence only for 0 < rho<1 Sep 26 at 16:24

With

 Limit[Integrate[ r^-2 BesselJ[0, r] D[BesselJ[0, r], {r, 2}], r]
r -> \[Infinity]]


$$\frac{20}{9 \pi }$$

and

    Series[Integrate[ r^-2 BesselJ[0, r] D[BesselJ[0, r], {r, 2}], r],
{r,   0, 2}]


$$\frac{1}{2 r}+\frac{5 r}{16}+O\left(r^3\right)$$

its obvious that the integral is finite for any interval with strictly positive lower bound

• +1. Also Series[r^-2 BesselJ[0, r] D[BesselJ[0, r], {r, 2}], {r, Infinity, 3}] // Normal results in -((2 Cos[\[Pi]/4 - r]^2)/(\[Pi] r^3)), confirming the convergence at infinity. Sep 26 at 10:41