# Harmonic function on annulus and finding Laurent series

1. Apr 25, 2014

### CAF123

1. The problem statement, all variables and given/known data
a)Find a harmonic function $u$ on the annulus $1< |z| < 2$ taking the value 2 in the circle $|z|=2$ and the value 1 in the circle $|z|=1$.

b)Determine all the isolated singularities of the function $f(z) = \frac{z+1}{z^3+4z^2+5z+2}$ and determine the residue at each one.

2. Relevant equations
Harmonic function satisfies Laplaces' equation

3. The attempt at a solution
a)I think I have to solve the Laplace equation $\partial^2_x u + \partial^2_y u = 0$ where u=u(x,y) with the boundary conditions $u|_{|z|=1} = 1$ and $u|_{|z|=2}=2$. $\partial^2_x u = - \partial^2_y u$. But how should I go about solving this?

b) First rewrite $f(z) = 1/(z+1)(z+2)$. I am trying to get this by constructing the suitable Laurent series about $z_o = -3/2$. In $|z+3/2| < 1,$ the function is analytic and so for |z+3/2| > 1, the function has a Laurent series. What is the easiest way to extract the Laurent series here? I am trying to rewrite f(z) in a form where on the demoninator I have 1-(1/(z+3/2)), so I can use the geometric series.

Many thanks.

2. Apr 26, 2014

### CAF123

Perhaps for the second question it would be better to expand about a different point within |z+3/2|<1?