The Fourier series of a complex function (,0 : (L, L) > C is defined as 00 $00 = Z cnei"""/L, where 1 L . cn = E L (p(x)eim1rx/LI (Notice the different signs in the two exponentials.) The doubly infinite sum can be interpreted as limem 2;:_N. Each c\" is complex in general. Recall that e'9 = cos 6 + i sin 6. (a) In the case where (000 is real-valued, the representation @(x) = 2:; Chaim\" is still valid. Thus all imaginary terms in the sum must cancel out. The way this happens is that enema/L + c_nei(\")'\""L is purely real for each n E {0, 1, 2, ...}. Denote the real and imaginary parts of c\" as c\" = (1,, + in. Determine how an and [3,, must relate to or-\" and [3." in order for Cneim'x'q' + c_,,e'i('\")\"x"" to be real. (b) Let @(x) be realvalued, so that c-" is determined by c" as in part (a). Show that the doubly infinite sum (.000 = 2:0 enema/L can be rewritten as an infinite sum @(x) = %A0 + 2:1[An cos(mrx/L) + .3n sin(mrx/L)], which you should recognize as the other standard form of a realvalued Fourier series. (c) Still assuming (p(x) is realvalued, give the formulas for A" and B\" in terms of an and [3,, that are implied by part (b). Show that these expressions for A\" and En coincide with the usual formulas for Fourier coefficients in terms of sines and cosines