(∫_(-Z)^Z)(1/(Sqrt[(X^2)+(Y^2)+(w^2)]))\[DifferentialD]w

Series[Log[1+x],{x,0,3}]

Table[{k,(LegendreP[2k+2,0]-LegendreP[2k,0])/(4k+3)},{k,0,5}]

Table[{j,Integrate[LegendreP[2j+1,x],{x,-1,0}]},{j,0,4}]

Table[{j,Integrate[LegendreP[2j+1,x],{x,0,1}]},{j,0,4}]

LegFcn0[j_]:= ((Sqrt[π])/(Gamma[j+1]Gamma[(1-2j)/2]))

(LegFcn0[j+1]-LegFcn0[j])/(4j+3)//FullSimplify

Table[{j,-((Sqrt[π])/(2 Gamma[(1/2)-j] Gamma[2+j]))},{j,0,4}]

eqns={(2/(2L+1))(A (a^L)+B (a^(-L-1)))==-V Ik,(2/(2L+1))(A (b^L)+B (b^(-L-1)))==V Ik}/.L->2j+1

Solve[eqns,{A,B}]

%//Simplify

eqns0={(2/(2L+1))(A (a^L)+B (a^(-L-1)))==V ,(2/(2L+1))(A (b^L)+B (b^(-L-1)))==V }/.{L->0 }

Solve[eqns0,{A,B}]

I1[j_]:=-((Sqrt[π])/(2 Gamma[(1/2)-j] Gamma[2+j]))

V=1;

Phi[r_,θ_,jmax_]:=(V=1;(( V)/(2 ))+Sum[(-((((a^(2+2 j))+(b^(2+2 j))) I1[j] (3+4 j) V)/(2 ((a^(3+4 j))-(b^(3+4 j))))) (r^(2j+1))+(((a^(2+2 j)) (b^(2+2 j)) ((a^(1+2 j))+(b^(1+2 j))) I1[j] (3+4 j) V)/(2 ((a^(3+4 j))-(b^(3+4 j)))))*(1/(r^(2j+2))))*LegendreP[2j+1,Cos[θ]],{j,0,jmax}])

Phi[r,θ,1]

Plot[Phi[1.1,θ,15]/.{a->1,b->1.2},{θ,0,π}]

Plot[Phi[1.1,θ,15]/.{a->0.01,b->1.2},{θ,0,π}]

Problem 3.2 stuff

(∫_(-1)^B)LegendreP[L,x]\[DifferentialD]x

Limit[%, L->0]

Table[{L,%%%},{L,1,5}]//Simplify

(LegendreP[2,Cos[x]]-1)/3  //Simplify

LegendreP[2,Cos[x]]