# How to plot a polar function with two variable?

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？

• Are you still looking for an answer?
– zhk
Mar 29 '17 at 10:28

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}
] • Thank you, but it seems that the result is not the same as desired. Is there any way to transform the expression to Cartesian coordinate and use ContourPlot ? Mar 16 '17 at 10:57

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}] • Thank you very much. I have replace R and phi to x and y by the converter formula directly. The result is correct now. Mar 17 '17 at 2:14