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

The course is about complex variables.

please show boundaries and in the complex coordinate system

COST f(x) := a - sin r on the interval -7 1 is real.2. (20 points) Recast the function f(x) from the previous problem as a (rational) function F(w) of the complex variable w := er, and extend F(w) to the entire complex plane. a. Find and classify all singular points w E C of the function F(w). b. Compute the (interior) Laurent series expansion of F(w) about the origin w = 0. What is the radius of convergence for this series? c. Use your result from part (b) to recompute the Fourier series expansion of f(x) from the previous problem. Confirm that your result here agrees with the one you found there