We can compute numerically:
In[]:=
f[rb_,l_,a_]:=NIntegrateExp
-
2
r
2
rb
r
2
(BesselJ[l,r])
,{r,0,a};
In[]:=
f[2,1,1/5]
Out[]=
0.0000986762
Compute a approximation about : a << 1 :
In[]:=
A=AsymptoticIntegrateExp
-
2
r
2
rb
r
2
(BesselJ[l,r])
,{r,0,a},{a,0,1},Assumptions->{l>0,rb>0}
Out[]=
1
2
2
a
-
2
a
2
rb

(
2
BesselJ[l,a]
-BesselJ[-1+l,a]BesselJ[1+l,a])
In[]:=
N[A/.rb->2/.l->1/.a->1/5,10]
Out[]=
0.00009834700891
Compute a approximation about : a ->Infinity:
AsymptoticIntegrateExp
-
2
r
2
rb
r
2
(BesselJ[l,r])
,{r,0,a},{a,Infinity,1},Assumptions->{l>0,rb>0}(*Can'tcompute.Weakness!!!*)
Out[]=
AsymptoticIntegrate
-
2
r
2
rb

r
2
BesselJ[l,r]
,{r,0,a},{a,∞,1},Assumptions{l>0,rb>0}
In[]:=
f[2,1,1000]
Out[]=
0.430539
In[]:=
N
1
2
2
rb
-
2
rb
2

2l
rb
-l
(
2
rb
)
BesselIl,
2
rb
2
-
-
2
a
2
rb

(1+Sin[2a-lπ])
aπ
/.rb->2/.l->1/.a->1000,6
Out[]=
0.430539