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.

(1)

Separation of variables gives radial powers

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

for positive angular modes and

1, lo g r

for the zero mode.

Table of contents
- Harmonic expansion on a disk Laplace equation in polar coordinates
- Harmonic extension in a circular sector Laplace equation in polar coordinates

Harmonic expansion on a disk ( $u (r, θ) = a_{0} + \sum_{n \geq 1} r^{n} (a_{n} cos (n θ) + b_{n} sin (n θ))$ )

 0  0

A real harmonic function on the unit disk has the displayed expansion, convergent locally uniformly with its derivatives in the interior. If its continuous boundary data have Fourier coefficients

A_{0}, A_{n}, B_{n}

in the convention

f (θ) = A_{0} /2 + \sum_{n \geq 1} (A_{n} cos (n θ) + B_{n} sin (n θ))

, its harmonic extension has

a_{0} = A_{0} /2

a_{n} = A_{n}

b_{n} = B_{n}

To obtain the radial factors, separation of variables in the Laplace equation in polar coordinates gives

r^{2} R^{''} + r R^{'} - λ R = 0

and

Θ^{''} + λ Θ = 0

. Angular periodicity restricts nonconstant modes to

λ = n^{2}

for positive integers

n

. The radial factors are

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

, with

1, lo g r

for the constant angular mode. Boundedness at the origin removes

r^{- n}

and

lo g r

. Each surviving factor

r^{n} cos (n θ)

r^{n} sin (n θ)

is a harmonic polynomial, the real or imaginary part of

(x + i y)^{n}

. The expansion can also be derived from the Poisson integral on the unit disk for the disk.

Harmonic extension in a circular sector ( $Φ (r, θ) = \sum_{n \geq 1} b_{n} (r / R)^{nπ / γ} sin (nπ θ / γ)$ )

 0  0

For

0 < θ < γ

and

0 < r < R

, a bounded harmonic function that vanishes on the straight edges and has outer boundary data

φ

has coefficients

b_{n} = \frac{2}{γ} \int_{0}^{γ} φ (θ) sin \frac{nπ θ}{γ} d θ .

(1)

This follows by separation of variables and Fourier orthogonality. Negative radial powers are excluded by boundedness at the vertex. Reflection-symmetric data

φ (θ) = φ (γ - θ)

have only odd Fourier modes. Knowing the resulting solution on the bisector determines all these odd coefficients by its radial power series. Without reflection symmetry, the even angular modes vanish on the bisector and cannot be recovered from that restriction alone.