8. Let ? be complex valued and piecewise continuous on ? D , and de?ne f : D ? C by 2 p f ( re i?…

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8.    Let ϕ be complex valued and piecewise continuous on ∂D, and define f : D → C by

f (reiθ )= 1 r

2π  0

Pr (θ

− t)ϕ(eit ) dt.

(a)     Prove that the functions Re f and Im f are harmonic.

(b)     Show that f ∈ H(D) if and only if

r 2π

eintϕ(eit) dt = 0,        n ∈ N.

0