Calculating $\int_{-\pi}^{+\pi} e^{ixt} e^{-i \omega t} dt$

Question

We know that Fourier Transform of $e^{ixt}$, where $x$ is a real parameter, $t\in \mathbb R$ is

$$\int_{-\infty}^{+\infty} e^{ixt} e^{-i \omega t} dt=\int_{-\infty}^{+\infty} e^{ixt-i \omega t} dt=\int_{-\infty}^{+\infty} e^{i(x- \omega )t} dt=2 \pi \delta(x- \omega)$$ The result is the same if you consider the following integral? $$\int_{-\pi}^{+\pi} e^{ixt} e^{-i \omega t} dt$$

Answer 1

$$ \color{#00f}{\large\mbox{No}}.\quad\mbox{The last one is equal to}\quad \left\lbrace\begin{array}{lcl} 2\,{\sin\left(\left[x - \omega\right]\pi\right) \over x - \omega} & \mbox{if} & x \not= \omega \\[3mm] 2\pi & \mbox{if} & x = \omega \end{array}\right. $$

Answer 2

No, for $x\neq \omega$ you have: $$\int_{-\pi}^{+\pi} e^{ixt} e^{-i \omega t} dt=2i\frac{e^{i\pi(x-\omega)}-e^{-i \pi(x-\omega)}}{2i(x-\omega)} =2i\frac{\sin(\pi (x-\omega))}{x-\omega} \neq 2 \pi \delta(x- \omega)$$