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

Chapter 3, Problem 3.68

(choose chapter or problem)

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(O) denote the length ofthe vector as a function of the angle (J. Then r( (J) is periodic in (J with period 27T and thus has a Fourier series representation. Let {ad denote the Fourier coefficients of r(O). (a) Consider now the projection x( 0) of the vector r( 0) onto the x-axis, as indicated in the figure. Determine the Fourier coefficients for x(O) in terms of the ak's. (b) Consider the sequence of coefficients bk = akejk7rt4. Sketch the figure in the plane that corresponds to this set of coefficients. (c) Repeat part (b) with (d) Sketch figures in the plane such that r(O) is not constant, but does have each of the following properties: (i) r(O) is even. (ii) The fundamental period of r(O) is 7T. (iii) The fundamental period of r(O) is 7T/2.