Suppose that $f(x)$ is periodic with period $[-\pi,\,\pi)$ and has the following complex Fourier coefficients:
$\ldots \quad c_0 = -3,\quad c_1 = 1-3i,\quad c_2 = 3+4i,\quad c_3 = -3i,\quad \ldots$

(A) Compute the following complex Fourier coefficients.
$c_{-3} =$ ,     $c_{-2} =$ ,     $c_{-1} =$

(B) Compute the real Fourier coefficients. (Remember that $e^{i\,kx} = \cos(k\,x) + i\,\sin(k\,x)$.)
$a_0 =$ $,\ a_1 =$ $,\ a_2 =$ $,\ a_3 =$ $,\ \ldots$
$\phantom{b_0 = X,}\qquad\qquad b_1 =$ $,\ b_2 =$ $,\ b_3 =$ $,\ \ldots$

(C) Compute the complex Fourier coefficients of the following.

      (i) The derivative $f'(x)$.
$c_{0} =$ ,     $c_{1} =$ ,     $c_{2} =$ ,     $c_{3} =$

      (ii) The shifted function $\displaystyle f\bigl(x+\frac{\pi}{6}\bigr)$
$c_{0} =$ ,     $c_{1} =$ ,
    $c_{2} =$ ,     $c_{3} =$

      (iii) The function $f(3x)$.
$c_{0} =$ ,     $c_{1} =$ ,     $c_{2} =$ ,     $c_{3} =$