f[c_,n_]=2(-Integrate[LegendreP[n,x],{x,0,c}]+Integrate[LegendreP[n,x],{x,c,1}] )
FindRoot[f[c,4],{c,.6}]

result: {c -> 0.654654}

g[n_]=f[c,n] /. %
a=Table[N[g[n] (2 n+1)/2],{n,0,10,2}]
phi[r_,c_]=Sum[a[[i]] r^(2 i-2) LegendreP[2 i-2,c],{i,1,6}]

ContourPlot[phi[Sqrt[x^2+z^2],z/Sqrt[x^2+z^2]], {x,-.9,.9},{z,-.9,.9},
ContourShading ->False,Contours -> Table[.05*i +phi[0,0],{i,-10,22}],
RegionFunction ->Function[{x, z}, x^2+z^2 <.9^2], PlotRangePadding->None]

phi2[r_,c_]=Sum[a[[i]] r^(2 i-2) LegendreP[2 i-2,c],{i,1,2}]

Plot[{phi[r,1],phi[r,0],phi2[r,1],phi2[r,0]},{r,0,.9},
PlotStyle -> {Directive[Black],Directive[Black],Directive[Red],Directive[Red]}]

