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Harmonic function on annulus and finding Laurent series

  1. Apr 25, 2014 #1

    CAF123

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    Gold Member

    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. jcsd
  3. Apr 26, 2014 #2

    CAF123

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