In[]:=
(********************************Functions*********************************)​​pn[θ_]:=Series[LegendreP[2*l+1,1,Cos[θ]],{l,Infinity,0},Assumptions->(0<θ<π&&l∈Integers&&l≥0)];​​SeriesCon[u_,θ_]:=Assuming[(0<θ<π)&&(l∈Integers)&&(l≥0)&&(
r
0
∈PositiveReals)&&(r∈PositiveReals),FullSimplify[Apply[And,SumConvergence[ReplaceAll[Apply[List,Expand[TrigToExp[ReplaceAll[FullSimplify[Normal[pn[θ]]],Sin[θ]->s]]]],s->Sin[θ]]*u,l]]]];​​(*************************************************************************)
Arθ
3
=
l
(-1)
-
5
2
-l
2
π
BesselJ
3
2
+2l,rγ(-1+2l)!!(3+4l)!!*
-1-2l
(γ
r
0
)
rγ
Gamma[2+l]
;​​SeriesCon[
Arθ
3
,θ]