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September 29

Solutions to Laplace's equation

My electromagnetism text derives a formula for azimuthally-symmetric solutions to Laplace's equation in spherical coordinates by separation of variables ie it first finds solutions of the form $V(r,\theta )=R(r)\Theta (\theta )$ . Plugging this into Laplace's equation, it gets two ODEs: ${\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=kR$ and ${\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=-k\Theta$ .

Now at this point, my text asserts that k is positive, without qualification or justification. Can anyone explain why k has to be positive? 74.15.136.9 (talk) 19:08, 29 September 2012 (UTC)[reply]

You could greatly help yourself specifying in which domain do you solve Laplace's equation, and with which boundary conditions. It is not easy to extract this information telepathically. Incnis Mrsi (talk) 19:44, 29 September 2012 (UTC)[reply]

Domain and boundary conditions are not being specified. The text is deriving a general formula for any azimuthally symmetric potential in spherical coordinates. The formula it gets is

V(r,\theta )=\sum _{l=0}^{\infty }\left(A_{l}r^{l}+{\frac {B_{l}}{r^{l+1}}}\right)P_{l}(\cos \theta )

, where

P_{l}

is the lth Legendre polynomial. — Preceding unsigned comment added by 74.15.136.9 (talk) 23:48, 29 September 2012 (UTC)[reply]

k is positive because the second equation needs to have periodic solutions for Θ. Sławomir Biały (talk) 00:49, 30 September 2012 (UTC)[reply]

Could you explain that in a little more detail please? 74.15.136.9 (talk) 01:56, 30 September 2012 (UTC)[reply]

Multiply both sides by

\Theta \sin ^{2}\theta

and integrate from zero to 2π. Integrate by parts. Periodicity implies the boundary terms cancel. Enforcing that both sides have the same sign forces k>0. Sławomir Biały (talk) 00:46, 1 October 2012 (UTC)[reply]

Legendre Polynomials

A Legendre polynomial $P_{n}(x)$ is supposed to satisfy the equation ${d \over dx}\left[(1-x^{2}){d \over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0$ . Substituting x = 1 into the above equation leads to the conclusion that $P_{n}(1)=0$ . But $P_{n}(x)=1$ . So what gives? 74.15.136.9 (talk) 19:44, 29 September 2012 (UTC)[reply]