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The piecewise function \( f(x) \) given in the image is defined as:

\[
f(x) = \begin{cases} 
-1, & -\pi < x < \frac{\pi}{2}, \\
1, & \frac{\pi}{2} < x < \pi.
\end{cases}
\]

To find the Fourier coefficients \( C_n \), we would express the function in its complex Fourier series form. The complex Fourier series for a periodic function \( f(x) \) is given by:

\[
f(x) = \sum_{n=-\infty}^{\infty} C_n e^{inx}
\]
where the coefficients \( C_n \) are determined by the formula:
\[
C_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx
\]

Let's compute \( C_n \) for the given piecewise function. Since the function is split into two intervals, we calculate the integral for each section separately.

\[
C_n = \frac{1}{2\pi} \left( \int_{-\pi}^{\frac{\pi}{2}} (-1) e^{-inx} \, dx + \int_{\frac{\pi}{2}}^{\pi} (1) e^{-inx} \, dx \right)
\]

I'll calculate the Fourier coefficients next.The Fourier coefficient \( C_n \) for the given piecewise function is:

\[
C_n = \begin{cases} 
\frac{i}{\pi n} \left( \cos\left(\frac{\pi n}{2}\right) - \cos(\pi n)\right), & n \neq 0, \\
-\frac{1}{2}, & n = 0.
\end{cases}
\]

This provides the general form of the Fourier coefficients for all \( n \).

Would you like more details on how this result was derived or help with further interpretation?

Here are some related questions to consider:
1. What is the significance of the Fourier series in signal processing?
2. How does the Fourier series approximation improve as more terms are added?
3. What happens if we change the intervals in the piecewise function?
4. How are the Fourier coefficients related to the frequency components of the function?
5. How does this compare to the Fourier series for even or odd functions?

**Tip**: For real-valued functions, the Fourier coefficients often have symmetries that simplify the calculation process.

This problem involves finding the Fourier coefficients Cn for a piecewise function using the complex Fourier series. The solution includes key mathematical concepts such as piecewise functions, complex exponentials, and the Fourier series formula. Suitable for undergraduate-level mathematics students.

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