How to plot a matrix in polar coordinates with color?

Hi I've been trying to plot this function in Matlab which is in polar coordinates:

V(rho, phi) = A * Sum(1/m * (rho/B)^m * sin(m*phi)) %m:1:1000:odd

A & B are constants and summation is on odd numbers from 1 to say 1000.

I wrote the following code to store a V for each rho and phi in a matrix. So the first column of this matrix is rho, the second one is phi and the third one is V. How can I plot rho and phi and represent V as the color so I'll have a 2D polar plot with color?

clc
clear all
v0=10; A=4.*v0./pi;k=1;
b=5; 
n=1:2:20;
for r=0.5:0.5:b
  for ph=0:0.2:2*pi
   V1=sum( A.*(1./n).*((r/b).^n).*sin(n*ph));
     M(k,1:3)=[r,ph,V1];
    k=k+1;
  end
end

Answer 1

[Andrew Newell]

I think most approaches will involve converting the coordinates to Cartesian. Here is one approach:

v0=10; A=4.*v0./pi;k=1;
b=5; 
n=1:2:20;
r = 0.5:0.5:b;
ph = 0:0.2:2*pi;
[r,ph] = meshgrid(r,ph);

V1 = zeros([size(r) numel(n)]);
for ii=1:numel(n)
  V1(:,:,ii) = A*(1/n(ii))*((r/b).^n(ii)).*sin(n(ii)*ph);
end
V1 = sum(V1,3);

contourf(r.*cos(ph),r.*sin(ph),V1)
axis square
colorbar

[A Hash]

Thank you. This is exactly what I was looking for.

[A Hash]

Quick question. How can I plot the gradient of V the same way?