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To find the Fourier Transform of the given function \( f(x) \), we start by defining the function mathematically:

\[
f(x) = \begin{cases} 
1 & \text{if } |x| < a \\
0 & \text{if } |x| > a 
\end{cases}
\]

### Step-by-Step Solution
The Fourier Transform \( F(k) \) of \( f(x) \) is given by:

\[
F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx
\]

Since \( f(x) \) is zero outside \( |x| < a \), we can limit the integration from \( -a \) to \( a \):

\[
F(k) = \int_{-a}^{a} 1 \cdot e^{-ikx} \, dx
\]

1. **Integrate \( e^{-ikx} \):**

   \[
   F(k) = \int_{-a}^{a} e^{-ikx} \, dx
   \]

   To integrate \( e^{-ikx} \), we use:

   \[
   \int e^{-ikx} \, dx = \frac{e^{-ikx}}{-ik} = -\frac{1}{ik} e^{-ikx}
   \]

2. **Evaluate the integral:**

   \[
   F(k) = \left[ -\frac{1}{ik} e^{-ikx} \right]_{-a}^{a}
   \]

   Substitute the limits:

   \[
   F(k) = -\frac{1}{ik} \left( e^{-ika} - e^{ika} \right)
   \]

3. **Simplify using Euler's formula:**

   \[
   F(k) = -\frac{1}{ik} \left( \cos(ka) - i \sin(ka) - \cos(ka) - i \sin(ka) \right)
   \]

   Notice that \( e^{-ika} - e^{ika} = -2i \sin(ka) \):

   \[
   F(k) = -\frac{1}{ik} (-2i \sin(ka))
   \]

   Simplify further:

   \[
   F(k) = \frac{2 \sin(ka)}{k}
   \]

So, the Fourier Transform of \( f(x) \) is:

\[
F(k) = \frac{2 \sin(ka)}{k}
\]

This is a form of the sinc function, which is common in Fourier Transforms of rectangular pulse functions.

---

Would you like additional details or have any questions? 

Here are some follow-up questions that might help expand your understanding:

1. How is the sinc function related to the Fourier Transform of rectangular functions?
2. What are the applications of Fourier Transforms in real-world signal processing?
3. How would the Fourier Transform change if \( f(x) \) had a different constant value instead of 1?
4. What is the significance of the result in terms of frequency components?
5. How does the parameter \( a \) influence the Fourier Transform \( F(k) \)?

**Tip:** Remember that the Fourier Transform of a rectangular function in the time domain corresponds to a sinc function in the frequency domain. This relationship is fundamental in signal processing and communications.

Find the Fourier Transform of the function f(x), where f(x) = 1 for |x| < a and 0 for |x| > a.

Learn how to compute the Fourier Transform of a rectangular pulse function f(x) with a step-by-step solution. This problem covers key mathematical concepts, including integration, Euler's formula, and the relationship between time-domain and frequency-domain representations.

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