Please write clearly and explain all the steps! Thank you!

The course is about complex variables.

1. (30 points) Consider the (real-valued) function COST f ( I) := a - sin r on the interval -T 1 is real. a. Extend f(x) to a function f(z) on the complex plane. Find and classify all singular points z E C of this extended function. That is, where are the singular points, and what types of singularities (simple poles, double poles, essential singularities, etc.) are they? b. Recall that the Fourier series coefficients for f(x) are defined by the integrals fn : = 27 f(x) e -inz dr. - IT Show that fn = 0, and that fin are complex conjugates of one another. c. Relate the Fourier coefficients fr for n > 0 to the closed contour integrals Jn ( R) = 2T JC(R) f(z) e-inz dz, where C(R) denotes the closed, rectangular contour in the complex plane with its corners at 2 = _ and z = +7 -iR. Why do we choose to close the rectangle in the lower half-plane? d. Evaluate fn by applying the residue theorem of Jn(R)