renegade05
Full Member
- Joined
- Sep 10, 2010
- Messages
- 260
\(\displaystyle ∇^2u = 0, \quad r > a\)
\(\displaystyle u = h(θ),\quad r = a,\)
Let \(\displaystyle v(ρ, θ)\) be given by Poisson’s integral formula, which is harmonic in the interior of the disk \(\displaystyle ρ < a\). Letting \(\displaystyle ρ = a^2/r\), show that \(\displaystyle u(r, θ) = v(a^2/r, θ)\) is harmonic in the exterior of the disk \(\displaystyle r > a\), then check values on \(\displaystyle r = a\).
I have for the Poisson's integral formula:
\(\displaystyle u(r,\theta) = \frac{r^2 -a^2}{2 \pi} \int_0 ^{2\pi} \frac{h(\phi)}{r^2+a^2-2ar\cos{(\theta-\phi)}}d\phi\)
How can I show this? I'm really at a loss with this one - I know the rule, show what you go. But I got nothing! I plugged in \(\displaystyle r \rightarrow ρ\) in the above equation to get \(\displaystyle v(ρ, θ)\) and then plugging in \(\displaystyle ρ = a^2/r\). But this is not going anywhere. Do I need to prove the laplacian is zero? That requires taking some nasty derivatives though, no? Not sure how to answer this one. Please help.
\(\displaystyle u = h(θ),\quad r = a,\)
Let \(\displaystyle v(ρ, θ)\) be given by Poisson’s integral formula, which is harmonic in the interior of the disk \(\displaystyle ρ < a\). Letting \(\displaystyle ρ = a^2/r\), show that \(\displaystyle u(r, θ) = v(a^2/r, θ)\) is harmonic in the exterior of the disk \(\displaystyle r > a\), then check values on \(\displaystyle r = a\).
I have for the Poisson's integral formula:
\(\displaystyle u(r,\theta) = \frac{r^2 -a^2}{2 \pi} \int_0 ^{2\pi} \frac{h(\phi)}{r^2+a^2-2ar\cos{(\theta-\phi)}}d\phi\)
How can I show this? I'm really at a loss with this one - I know the rule, show what you go. But I got nothing! I plugged in \(\displaystyle r \rightarrow ρ\) in the above equation to get \(\displaystyle v(ρ, θ)\) and then plugging in \(\displaystyle ρ = a^2/r\). But this is not going anywhere. Do I need to prove the laplacian is zero? That requires taking some nasty derivatives though, no? Not sure how to answer this one. Please help.