In two-dimensional polar coordinates, the Helmholtz differential equation is
(1)
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Attempt separation of variables by writing
(2)
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then the Helmholtz differential equation becomes
(3)
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Multiply both sides by to obtain
(4)
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The solution to the second part of (4) must be periodic, so the differential equation is
(5)
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which has solutions
(6)
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(7)
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This has solution
(8)
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where and are Bessel functions of the first and second kinds, respectively. Combining the solutions gives the general solution
(9)
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