Applying Jensen's formula to polynomials? Prove that $$\frac{1}{2\pi}\int_{0}^{2\pi}|f(e^{i\theta})|^2d\theta=\sum_{k=0}^n|c_k|^2$$ for each polynomial $f(z)=\sum_{k=0}^nc_kz^k$.
The hint given by the homework is: show first that for integer $j$, $k$, $$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{e^{ik\theta}}{e^{ij\theta}}d\theta=1 \text{ for } j=k, =0 \text{ for } j\neq k.$$
I am completely lost even when the hint is given. How should I proceed to approach this problem?

To prove the identity given in the hint, note that when $j=k$, you're just integrating a constant function $1$. For $j \neq k$, you can use the fundamental theorem of calculus to compute the integral, and periodicity of $e^{i(k-j)\theta}$ gives that the integral is zero.
Now to complete the problem, write
$$ |f(e^{i\theta})|^2 = f(e^{i\theta})\overline{f(e^{i\theta})} = \sum_{k=0}^n\sum_{\ell = 0}^n c_k\overline{c_\ell} \frac{e^{ik\theta}}{e^{i\ell\theta}}$$.
When you integrate this using the result of the hint, you'll get what you wanted. 
