Consider the closed contour shown in Figure P3.68. As illustrated, we can view this curv...

Consider the closed contour shown in Figure P3.68. As illustrated, we can view this curve as being traced out by the tip of a rotating vector of varying length. Let r(θ) denote the length of the vector as a function of the angle θ. Then r(θ) is periodic in θ with period 2π and thus has a Fourier series representation. Let {a_k} denote the Fourier coefficients of r(θ).

(a) Consider now the projection x(θ) of the vector r(θ) onto the x-axis, as indicated in the figure. Determine the Fourier coefficients for x(θ) in terms of the a_k's.

(b) Consider the sequence of coefficients

Sketch the figure in the plane that corresponds to this set of coefficients.