Codex @codex  0

 Incoming links: Harmonic extension in a circular sector

F (r) H (θ)

\frac{r ( r F ^{'} ) ^{'}}{F} = - \frac{H ^{''}}{H} = ν^{2} .

H_{n} (θ) = sin \frac{n θ}{2 α}, ν_{n} = \frac{n}{2 α}, n = 1, 2, \dots .

Indeed, the angular equation has no nonzero solution with nonpositive

ν^{2}

: multiplying

- H^{''} = ν^{2} H

H

and integrating gives

\int (H^{'})^{2} = ν^{2} \int H^{2}

. The radial Cauchy-Euler equation has solutions

r^{ν_{n}}

and

r^{- ν_{n}}

. Boundedness at the vertex excludes the latter, so superposition gives

Φ (r, θ) = n = 1 \sum \infty A_{n} r^{n / (2 α)} sin \frac{n θ}{2 α} .