(a) Let f (z) denote a function which is analytic in some annular domain about the origin that includes the unit circle z = ei( (ˆ’Ï€ ‰¤ ( ‰¤ Ï€). By taking that circle as the path of integration in expressions (2) and (3), Sec. 60, for the coefficients an and bn in a Laurent series in powers of z, show that

When z is any point in the annular domain.
(b) Write u(θ) = Re[f (eiθ)] and show how it follows from the expansion in part (a) that

This is one form of the Fourier series expansion of the real-valued function u(θ) on the interval ˆ’Ï€ ‰¤ θ ‰¤ Ï€. The restriction on u(θ) is more severe than is necessary in order for it to be represented by a Fourier series.ˆ—

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