[8] Find the complex Fourier integral representation of f(x) = e2x 0 |x| < 1, |x ≥ 1.
[8] Find the complex Fourier integral representation of f(x) = e2x 0 |x| < 1, |x ≥ 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
**[8]** Find the complex Fourier integral representation of \( f(x) \) defined as:
\[
f(x) =
\begin{cases}
e^{2x} & |x| < 1, \\
0 & |x| \geq 1.
\end{cases}
\]
### Detailed Explanation
For this problem, we are asked to determine the complex Fourier integral representation of a given piecewise function \( f(x) \).
#### Function Details:
- When \( |x| < 1 \), the function \( f(x) \) is defined as \( e^{2x} \).
- When \( |x| \geq 1 \), the function \( f(x) \) is defined as \( 0 \).
This task involves deriving a form of the function \( f(x) \) that is expressed as an integral involving complex exponentials. This representation allows analyzing the frequencies that compose the function \( f(x) \).
To achieve this, we typically apply principles from Fourier analysis.
### Steps to Approach the Problem:
1. **Express f(x) in terms of Complex Fourier Transform:**
\[
\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx
\]
2. **Evaluate integral within the defined intervals:**
Since \( f(x) \) is non-zero only for \( |x| < 1 \):
\[
\hat{f}(k) = \int_{-1}^{1} e^{2x} e^{-ikx} \, dx = \int_{-1}^{1} e^{(2-ik)x} \, dx
\]
3. **Integrate using the appropriate integration techniques.**
4. **Simplify the resulting expression to write \( f(x) \) as the inverse Fourier transform:**
\[
f(x) = \int_{-\infty}^{\infty} \hat{f}(k) e^{ikx} \, dk
\]
In a formal course or textbook, each of these steps would be elaborated upon, with specific techniques for solving the integrals being provided.
This process highlights the transformation of a time or spatially defined function into a form that reveals its frequency components, a foundational concept in signal processing and harmonic](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2Fa2c00922-c862-4bcd-b7f8-38fafbdf2f13%2Fkfx897_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**[8]** Find the complex Fourier integral representation of \( f(x) \) defined as:
\[
f(x) =
\begin{cases}
e^{2x} & |x| < 1, \\
0 & |x| \geq 1.
\end{cases}
\]
### Detailed Explanation
For this problem, we are asked to determine the complex Fourier integral representation of a given piecewise function \( f(x) \).
#### Function Details:
- When \( |x| < 1 \), the function \( f(x) \) is defined as \( e^{2x} \).
- When \( |x| \geq 1 \), the function \( f(x) \) is defined as \( 0 \).
This task involves deriving a form of the function \( f(x) \) that is expressed as an integral involving complex exponentials. This representation allows analyzing the frequencies that compose the function \( f(x) \).
To achieve this, we typically apply principles from Fourier analysis.
### Steps to Approach the Problem:
1. **Express f(x) in terms of Complex Fourier Transform:**
\[
\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx
\]
2. **Evaluate integral within the defined intervals:**
Since \( f(x) \) is non-zero only for \( |x| < 1 \):
\[
\hat{f}(k) = \int_{-1}^{1} e^{2x} e^{-ikx} \, dx = \int_{-1}^{1} e^{(2-ik)x} \, dx
\]
3. **Integrate using the appropriate integration techniques.**
4. **Simplify the resulting expression to write \( f(x) \) as the inverse Fourier transform:**
\[
f(x) = \int_{-\infty}^{\infty} \hat{f}(k) e^{ikx} \, dk
\]
In a formal course or textbook, each of these steps would be elaborated upon, with specific techniques for solving the integrals being provided.
This process highlights the transformation of a time or spatially defined function into a form that reveals its frequency components, a foundational concept in signal processing and harmonic
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