How to plot a polar function with two variable?

Question

The expression in polar coordinate is

LGo[l_, m_, V_, R_, ϕ_] := R^l LaguerreL[m - 1, l, V R^2] Exp[-1/2 V R^2] Cos[l ϕ];

where l,m,V are parameters, and R,phi are variable. However, It can't be visualized by Polar Plot.

Is there any way to show it？

Answer 1

There are several ways. One easy solution is to create a Table for several radii and plot them or use them in a Manipulate

LGo[l_, m_, V_, R_, ϕ_] := 
  R^l LaguerreL[m - 1, l, V R^2] Exp[-1/2 V R^2] Cos[l ϕ];

Manipulate[
 With[{expr = Table[LGo[l, m, V, r, phi], {r, 0.1, 1, 0.1}]},
  PolarPlot[expr, {phi, 0, 2 Pi}]
  ],
 {l, 1, 2},
 {m, 1, 2},
 {V, 1, 2}
 ]

Answer 2

In response to OP's comment,

Is there any way to transform the expression to Cartesian coordinate and use ContourPlot?

R1 = Sqrt[x^2 + y^2];
ϕ = ArcTan[y/x];
LGo[l_, m_, V_, R_, ϕ_] := R1^l LaguerreL[m - 1, l, V R1^2] Exp[-1/2 V R1^2] Cos[l ϕ];
Manipulate[ ContourPlot[LGo[l, m, V, x, y], {x, -5, 5}, {y, -5, 5}, 
                        PerformanceGoal -> "Quality"], {l, 1, 2}, {m, 1, 2}, {V, 1, 2}]