Let cm be the m-th coefficient in the complex exponential form of the Fourier series for a 27t-periodic real-valued function f (#). 1. Show that C-m = Cm- 2. If f has even symmetry, show that Cm is real. 3. If f has odd symmetry, show that Cm is imaginary. 4. Let o be a real number, and consider the phase-shifted function g (t) = f (t - q). Show that the Fourier coefficients g,, for g are g, = CmeTime 5. You will show using three different methods that the Fourier coefficients d,, for f' (t) are d, = imcm: (a) Differentiate term-by-term the series for f (1). (b) Use integration by parts to evaluate the integral d, = 2 ( f' (t) e int dt. Be sure to explain how you handle the boundary term that arises from IBP. (c) Use the result of part 4 to evaluate lim,_, ( 9-f(Q 6. In part 5 you showed that the coefficients for f' (1) are dy, - imc,,. Here, we'll suppose we're given the coefficients d,, and wish to reconstruct the coefficients Cm for the series for f (t). (a) Notice that when m = 0, we have do - 0 for any co. Naively solving for the coefficient co to reconstruct f, we find co = 0/0. Explain why it makes sense that it's impossible to determine co uniquely, and why it's legitimate to choose any arbitrary value for co. (b) Suppose the series for f' (t) converges in L'. Show that the series for f (t) will also con- verge in LZ (c) Suppose the series for f' (1) passes the Weierstrass M-test. Show that the series for f (!) will also pass the M-test