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 Incoming links: Axisymmetric harmonic interpolation between concentric spheres

Φ = R (r) P (θ)

and multiply the Laplace equation by

r^{2} / (RP)

\frac{( r ^{2} R ^{'} ) ^{'}}{R} = - \frac{( s i n θ P ^{'} ) ^{'}}{s i n θP} = λ .

With

ξ = cos θ

, the angular equation is

[(1 - ξ^{2}) P_{ξ}]^{'} + λ P = 0

. Regularity at both poles selects the Legendre polynomials

P_{n}

and

λ = n (n + 1)

. The radial Cauchy-Euler equation has solutions

r^{n}, r^{- n - 1}

. Expanding the angular data in this complete orthogonal system therefore gives

Φ (r, θ) = \sum_{n = 0}^{\infty} (A_{n} r^{n} + B_{n} r^{- n - 1}) P_{n} (cos θ),

with the usual convergence required of a harmonic expansion on the region considered. Angular Legendre functions singular at the poles are excluded.

For

0 < a < b

, the outer data have expansion

b^{2} cos^{2} θ = b^{2} /3 + (2 b^{2} /3) P_{2} (cos θ)

, while the inner data are pure

P_{0}

. Thus only degrees zero and two are needed. A radial interpolation matching each pair of boundary values gives

Φ (r, θ) = \frac{a ^{3} ( b - r ) + ( b ^{3} /3 ) ( r - a )}{r ( b - a )} + \frac{2 b ^{2}}{3} \frac{r ^{2} - a ^{5} / r ^{3}}{b ^{2} - a ^{5} / b ^{3}} P_{2} (cos θ) .

1, r^{- 1}

and takes values

a^{2}, b^{2} /3

at the two radii. The second is a combination of

r^{2}, r^{- 3}

and takes values zero and

2 b^{2} /3