Φ  =  
      Log[(ρ^2 + r^2 - 2  ρ r Cos[ϕ])/(ρ^2 + 
              R^2 - 2  ρ R Cos[ϕ])]

Series[Φ, {ρ, Infinity, 4}]

f[x_]  =  (x^2 - 1)/(x^2 + 1 - 2 x Cos[ϕ])

Plot[{f[2], f[4]}, {ϕ, 0, 2 π}, 
      AxesLabel ->  {"ϕ(radians)", ""}, 
      PlotStyle ->  {Thickness[0.01], {Thickness[0.01], 
            Dashing[{0.05, 0.03}]}}, 
      TextStyle -> {FontFamily -> "Times", 
          FontSize -> 12}, 
      PlotLabel ->  "σ [solid R/b=2; dashed R/b=4]"]

Export["prob211.eps", %24]