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Laplace equation in polar coordinates

In plane polar coordinates, Laplace's equation is

\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial ϕ}{\partial r}) + \frac{1}{r ^{2}} \frac{\partial ^{2} ϕ}{\partial θ ^{2}} = 0.

Separation of variables gives radial powers

r^{n}, r^{- n}

for positive angular modes and

1, lo g r

for the zero mode.

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