In[]:=
Arθ
1
=
l+1
(-1)
*LegendreP[1+2l,1,Cos[θ]]*(-1+2l)!!*
1+2l
r
r
0
2(2+2l)!!
;​​
Arθ
2
=
l+1
(-1)
*LegendreP[1+2l,1,Cos[θ]]*(-1+2l)!!*
-2-2l
r
r
0
2(2+2l)!!
;​​
B
r
1
=
l
(-1)
(1+l)(1+2l)(-1+2l)!!LegendreP[1+2l,Cos[θ]]
2l
r
r
0
(2+2l)!!
r
0
;​​
B
r
2
=
l
(-1)
(1+l)(1+2l)(-1+2l)!!LegendreP[1+2l,Cos[θ]]
-2(1+l)
r
r
0
r(2+2l)!!
;​​
Arθ
3
=
l
(-1)
-
5
2
-l
2
π
BesselJ
3
2
+2l,rγ(-1+2l)!!(3+4l)!!LegendreP[1+2l,1,Cos[θ]]
-1-2l
(γ
r
0
)
rγ
Gamma[2+l]
;​​
Arθ
4
=-
l
(-1)
-
5
2
-l
2
π
BesselY
3
2
+2l,rγ(-1+2l)!!LegendreP[1+2l,1,Cos[θ]]
2(1+l)
(γ
r
0
)
rγ
(1+4l)!!Gamma[2+l]
;​​​​
B
r
3
=
l
(-1)
-
3
2
-l
2
(1+2l)
π
γBesselJ
3
2
+2l,rγ(-1+2l)!!(3+4l)!!LegendreP[1+2l,Cos[θ]]
-1-2l
(γ
r
0
)
3/2
(rγ)
l!
;​​
B
r
4
=-
l
(-1)
-
3
2
-l
2
(1+2l)
π
γBesselY
3
2
+2l,rγ(-1+2l)!!LegendreP[1+2l,Cos[θ]]
2(1+l)
(γ
r
0
)
3/2
(rγ)
l!(1+4l)!!
;​​​​
B
z
1
=
l
(-1)
-
3
2
-l
2
(1+2l)
π
γBesselJ
3
2
+2l,zγ(-1+2l)!!(3+4l)!!
-1-2l
(γ
r
0
)
3/2
(zγ)
Gamma[1+l]
;​​​​
B
z
2
=-
l
(-1)
-
3
2
-l
2
(1+l)(1+2l)
π
γBesselY
3
2
+2l,zγ(-1+2l)!!
2(1+l)
(γ
r
0
)
3/2
(zγ)
(1+4l)!!Gamma[2+l]
;
In[]:=
SumConvergence[{
Arθ
1
,
Arθ
2
,
B
r
1
,
B
r
2
,
Arθ
3
,
Arθ
4
,
B
r
3
,
B
r
4
,
B
z
1
,
B
z
2
},l,Method->"RatioTest",Assumptions->(0<θ<π)&&(l∈PositiveIntegers)&&(
r
0
∈PositiveReals)&&(r∈PositiveReals)]
Out[]=
$Aborted
In[]:=
SumConvergence[
n
t
*LegendreP[n,Cos[θ]],n,Method->"RatioTest",Assumptions->(0<θ<π)]
Out[]=
SumConvergence[
n
t
LegendreP[n,Cos[θ]],n,MethodRatioTest,Assumptions0<θ<π]