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
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2 Answers
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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
answered Sep 26, 2023 at 7:52
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$\begingroup$ Does this mean that the integral is convergent in
0 < rho < infinity? $\endgroup$ Commented Sep 26, 2023 at 15:21 -
1$\begingroup$ No, convergence only for
0 < rho<1$\endgroup$ Commented Sep 26, 2023 at 16:24
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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
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$\begingroup$ +1. Also
Series[r^-2 BesselJ[0, r] D[BesselJ[0, r], {r, 2}], {r, Infinity, 3}] // Normalresults in-((2 Cos[\[Pi]/4 - r]^2)/(\[Pi] r^3)), confirming the convergence at infinity. $\endgroup$ Commented Sep 26, 2023 at 10:41
In[29]:= FunctionPoles[ii, r] Out[29]= {{0, 2}}This shows that the integration path cannot contain the origin. $\endgroup$