3. De?ne ? : ? D ? R by ? ( z )= ( 1 if Im z = 0 , 0 if Im z 0 . Find the solution to the…

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3.    Define ϕ : ∂D → R by

 

ϕ(z)=

(1    if Im z ≥ 0,

0   if Im z 0.

Find the solution to the Dirichlet problem on D for the boundary function ϕ by directly calculating (6.2.6) and taking the real part. (Hint: Separate into two integrals and be careful to account for branches of the logarithm.)