Fourier coefficient problem

Question

Calculate fourier coefficient $\hat{s}(-1)$, where 1-periodic signal $s$ :$\Bbb{R}/\Bbb{Z}\to\Bbb{C}$ is defined with equation $s(t)=(2cos(\pi t))^{16}$

Answer 1

The general formula for a $f$ with period L is: \begin{align*} \hat{f}(j)= \frac{1}{L} \int_c^{c+L} e^{-\frac{2\pi i j }{L}t}f(t)dt \end{align*} where $c \in \mathbb{R}$ arbitrary
and therefore $\hat{s}(-1)= \frac{1}{1} \int_0^{0+1} e^{-\frac{2\pi i (-1) }{1}t}(2cos(\pi t))^{16}dt = 2^{16}\int_0^{1} e^{2\pi it}cos(\pi t)^{16}dt \overbrace{=}^{Wolfram|Alpha}$

$\frac{1}{126 \pi} (1441440 \pi t+504504 i e^{-2 i \pi t}-810810 i e^{2 i \pi t}+137592 i e^{-4 i \pi t}-360360 i e^{4 i \pi t}+38220 i e^{-6 i \pi t}-168168 i e^{6 i \pi t}+8820 i e^{-8 i \pi t}-68796 i e^{8 i \pi t}+1512 i e^{-10 i \pi t}-22932 i e^{10 i \pi t}+168 i e^{-12 i \pi t}-5880 i e^{12 i \pi t}+9 i e^{-14 i \pi t}-1080 i e^{14 i \pi t}-126 i e^{16 i \pi t}-7 i e^{18 i \pi t})|_0^1$

$\overbrace{=}^{Mathematica} 11440$