The given image appears to describe a piecewise-defined function $f(x)$ and asks to calculate its Fourier transform ($\mathcal{F}(f(x))$).

Steps to Compute the Fourier Transform:

1. Piecewise Definition: From the image:

   \begin{cases} e^{2i\pi x}, & -1 < x < 1, \\ 0, & \text{otherwise.} \end{cases}$$

2. Fourier Transform Definition: The Fourier transform is generally defined as: $F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx.$ Since $f(x) = 0$ outside the interval $(-1, 1)$, the integral reduces to: $F(\omega) = \int_{-1}^{1} e^{2i\pi x} e^{-i\omega x} dx = \int_{-1}^{1} e^{(2i\pi - i\omega)x} dx.$

3. Simplify the Exponential: Combine terms in the exponent: $F(\omega) = \int_{-1}^{1} e^{i(2\pi - \omega)x} dx.$

4. Integration: For $\int e^{i k x} dx$, the result is: $\int e^{i k x} dx = \frac{1}{i k} e^{i k x}, \, \text{where } k \neq 0.$ Here, $k = 2\pi - \omega$, so: $F(\omega) = \left[ \frac{e^{i(2\pi - \omega)x}}{i(2\pi - \omega)} \right]_{-1}^{1}.$

5. Evaluate the Limits: Substitute $x = 1$ and $x = -1$: $F(\omega) = \frac{e^{i(2\pi - \omega)} - e^{-i(2\pi - \omega)}}{i(2\pi - \omega)}.$

6. Simplify Using Sine: Use the identity $e^{i\theta} - e^{-i\theta} = 2i \sin(\theta)$: $F(\omega) = \frac{2i \sin(2\pi - \omega)}{i(2\pi - \omega)} = \frac{2 \sin(2\pi - \omega)}{2\pi - \omega}.$

   Finally: $F(\omega) = \frac{\sin(2\pi - \omega)}{2\pi - \omega}.$

Final Fourier Transform:

$F(\omega) = \frac{\sin(2\pi - \omega)}{2\pi - \omega}.$

Let me know if you want detailed steps or clarifications!
Here are 5 questions to consider next:

1. What are the properties of the Fourier transform of even or odd functions?
2. How does scaling the function in time affect its Fourier transform?
3. What happens to the Fourier transform when $f(x)$ is shifted in time?
4. How is the sinc function related to the Fourier transform of a rectangular signal?
5. Can the Fourier transform be extended to multidimensional signals?

Tip: Always check the limits of integration when calculating Fourier transforms for piecewise functions.