> [!NOTE] > A real [[Trigonometric Polynomials|trigonometric polynomial]] can be rewritten as a complex trig polynomial and vice versa. ###### Proof: From [[Exponential Function in Terms of Trigonometric Functions (Euler's Formula)]], $\cos(\theta)= \frac{1}{2} (e^{i\theta}+e^{-i\theta}) \quad \text{ and } \sin (\theta)=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})$hence we can easily transform the real version into the complex version. $a_{0}+ \sum_{k=1}^{n} (a_{k}\cos(kx)+b_{k}\sin(kx))= \sum_{k=-n}^n c_{k}e^{ikx} \quad \text{ where } c_{k} = \begin{cases} \frac{1}{2} (a_{k} - ib_{k}), & k>0 \\ a_{0}, & k=0, \\ \frac{1}{2}(a_{-k}+ib_{-k}). & k<0. \end{cases} $noting that $c_{-k}=\overline{c_{k}}$. Vice versa, we can choose unique $a_{k}$ and $b_{k}$ such that $a_{0}=c_{0}, \quad a_{k}=c_{k}+c_{-k}, \quad b_{k}=i(c_{k}-c_{-k})$