Temperature distribution on a circular plate

[zi-lao-lan]

Hi, I have a maths exam tommorrow (!) and I'm stuck on something:

Its about solving LaPlace's equation in plane polar coordinates. There is a circular plate witha temp dist that satisfies

(1/r)(d/dr(r.df/dr)) + 1/r² . d²/dp² = 0
(p is phi)
so I used a separable eq of the form f=RP

and solved it to get R = Cr^m+Dr^-m
when m doesn't equal 0

and R(r) = A + B ln(r)
when m=0

The question is:
What boundary condition does R(r) satisfy at r = 0? Use this to show that
the general solution to this problem is

f= A<sub>0 + SUM OF(r^m)( Am cos(mp) + Bm sin(mp))

the explanation is because you can't have negative powers of m when R(r) is finite at the centre. Why can't you have negative powers of m? Negative powers will give fractions but they will still be finite, so why isn't it

f= A0 + SUM OF(Cmr^m+Dmr^-m)( Am cos(mp) + Bm sin(mp))

Thanks if you can help before tommorrow!</sub>

 

[jambaugh]

Negative powers of r not m. The negative powers of r gives:

$$R(0) = (0)^{-m}$$
with m positive.
BOOM! It blows up at 0.

 

[zi-lao-lan]

Oh
Thats really obvious, sorry