In 2D Polar Coordinates, attempt Separation of Variables by writing

(1) 
then the Helmholtz Differential Equation becomes

(2) 
Divide both sides by

(3) 
The solution to the second part of (3) must be periodic, so the differential equation is

(4) 
which has solutions
Plug (4) back into (3)

(6) 
This is an Euler Differential Equation with
and
. The roots are .
So for , and the solution is

(7) 
But since blows up at , the only possible physical solution is . When , , so

(8) 
But since blows up at , the only possible physical solution is
. The solution for is
then

(9) 
for , 1, ...and the general solution is

(10) 
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGrawHill, pp. 502504, 1953.
© 19969 Eric W. Weisstein
19990525