4. Let ? be a real-valued piecewise continuous function on ? D . If ? ( e i? ) = ? ( e -i? ) for…

4.    Let ϕ be a real-valued piecewise continuous function on ∂D. If ϕ(eiθ ) = ϕ(e−iθ ) for all θ ∈ R, prove that ϕ has a Fourier series of the form

 

∼ α   ∞

ϕ(eiθ )          0  +2      αn

2

n=1

cos nθ,

where the Fourier coefficients are all real.

 

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