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Interactive Audio Lesson
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Fourier Sine Integral Representation
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Let's explore the Fourier sine integral representation using the function defined as f(x) = 1 for 0 < x < a and 0 for x ≥ a.
Why do we use the sine integral for this function?
Good question! We use the sine integral because the function is odd over the interval from 0 to infinity, making it suitable for sine representation.
How do we calculate B(ω)?
We calculate B(ω) with the formula that involves integrating sine. Remember: B(ω) = (1/π) ∫ from 0 to a of sin(ωt) dt. Do you recall how to perform this integral?
I think we can use the formula for the integral of sin. It leads us to B(ω) being (1 - cos(ωa)) / (πω).
Exactly! Now, can you put that back into the integral for f(x)?
So, we would integrate sin(ωx) multiplied by B(ω)?
Yes! And this leads us to the final result. Remember, practice makes perfect!
Today, we've used the Fourier sine integral to represent a function. Next, let's shift our focus to cosine integrals.
Fourier Cosine Integral
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Now, let's discuss the Fourier cosine integral using the function f(x) = e^{-ax} for x > 0, a > 0.
Why is this function suitable for a cosine integral?
Good observation! This function is even, which allows us to use the cosine integral representation. We define A(ω) for this case.
How do we find A(ω)?
To find A(ω), we integrate e^{-at} cos(ωt) dt from 0 to infinity. The known result is crucial here, which gives us the formula.
I remember that the integral yields A(ω) = 2a / (π(a² + ω²)).
Right! Now place this A(ω) back into the cosine integral and simplify.
So we end up with the Fourier cosine integral for f(x)?
Correct! This illustrates another powerful application of the Fourier integral. Key takeaway: Understand the nature of your function to choose the right approach.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section provides two worked examples: the first involves calculating the Fourier sine integral of a step function, while the second covers the Fourier cosine integral for an exponential function. These examples help consolidate the theoretical concepts of Fourier integrals in practical scenarios.
Detailed
Worked Examples
This section focuses on practical applications of Fourier integrals through worked examples to better illustrate key concepts from earlier sections. Two examples are provided:
Example 1: Fourier Sine Integral Representation
In the first example, we evaluate the Fourier sine integral representation of a function defined as:
- f(x) = 1 for 0 < x < a
- f(x) = 0 for x ≥ a
Since the function is odd over the interval (0, ∞), the Fourier sine integral is applicable:
$$
\int_{0}^{\infty} f(x) = \int_{0}^{\infty} B(\omega) \sin(\omega x) d\omega\n$$
Where B(ω) is computed as:
$$
B(\omega) = \frac{1}{\pi} \int_{0}^{a} \sin(\omega t) dt = \frac{1 - \cos(\omega a)}{\pi \omega}
$$
This leads to:
$$
\int_{0}^{\infty} f(x) = \frac{1 - \cos(\omega a)}{\pi \omega} \sin(\omega x) d\omega
$$
Example 2: Fourier Cosine Integral
The second example finds the Fourier cosine integral for the function f(x) = e^{-ax}, with a > 0 for x > 0. Here, we apply:
$$
\int_{0}^{\infty} f(x) = \int_{0}^{\infty} A(\omega) \cos(\omega x) d\omega
$$
Calculating A(ω):
$$
A(\omega) = \frac{2}{\pi} \int_{0}^{\infty} e^{-at} \cos(\omega t) dt = \frac{2a}{\pi (a^{2}+\omega^{2})}
$$
Thus, the final Fourier cosine integral representation verifies that:
$$
\int_{0}^{\infty} f(x) = \frac{2a}{\pi} \int_{0}^{\infty} \frac{\cos(\omega x)}{(a^{2}+\omega^{2})} d\omega
$$
Both examples demonstrate the practical use of Fourier integrals, reflecting complex functions in terms of sine and cosine expressions, reinforcing the application of theoretical concepts during problem-solving.
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Example 1: Fourier Sine Integral of a Step Function
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Evaluate the Fourier sine integral representation of the function:
$$
f(x) = \begin{cases} 1, & 0 < x < a \\ 0, & x \geq a \end{cases}$$
Solution:
Since f(x) is odd over (0, ∞), use the sine integral:
$$
\int_0^{\infty} f(x) = B(\omega) \sin(\omega x) d\omega
$$
Where:
$$
B(\omega) = \frac{1}{\pi} \int_0^{a} \sin(\omega t) dt = \frac{1}{\pi \omega} [1 - \cos(\omega a)]
$$
Thus,
$$
\int_0^{\infty} f(x) = \frac{1 - \cos(\omega a)}{\pi \omega} \sin(\omega x) d\omega
$$
Detailed Explanation
In this example, we are evaluating the Fourier sine integral for a piecewise function defined in the interval (0, a). The function has a value of 1 between 0 and a, and 0 elsewhere. The function is classified as odd over the positive interval, which lets us apply the sine integral formula. The formula involves integrating the sine function multiplied by a coefficient, leading to a representation that combines both sine and cosine functions. The expression for B(ω) includes the use of the sine integral to represent the contribution from the defined range.
Examples & Analogies
Imagine you are looking at a sensor that only detects activity in a room, active only when people are present. If you think of the area up to 'a' as the room, and activity as the value of the function, outside the room (i.e., greater than 'a'), there's no detection (the function is zero). The Fourier sine integral helps us observe how the activity oscillates and contributes to different frequencies, similar to how the sensor might respond to varying levels of noise or movement.
Example 2: Fourier Cosine Integral of an Exponential Decay Function
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Find the Fourier cosine integral of $$f(x) = e^{-ax}, \; a > 0, \; \text{for } x > 0.$$
Solution:
Use:
$$
\int_0^{\infty} f(x) = A(\omega) \cos(\omega x) d\omega
$$
Where:
$$
A(\omega) = \frac{2}{\pi} \int_0^{\infty} e^{-at} \cos(\omega t) dt
$$
Using a known integral:
$$
\int_0^{\infty} e^{-at} \cos(\omega t) dt = \frac{a}{a^2 + \omega^2}
$$
Hence,
$$
\int_0^{\infty} f(x) = \frac{2a}{\pi (a^2 + \omega^2)} \cos(\omega x) d\omega
$$
This verifies the cosine integral representation of $$f(x) = e^{-ax}$$.
Detailed Explanation
In this example, we aim to find the Fourier cosine integral for the exponential decay function defined over positive values. The function's nature being defined indicates it covers all values exponentially diminishing towards zero. To compute the integral, we used the formula that connects the decay function with oscillatory components using a known integral form. The relationship outlined leads to a simplified result that correlates decay with frequency, showcasing how the function contributes to different cosine frequencies.
Examples & Analogies
Think of a diminishing sound from a speaker that fades out over time. The exponential decay represents how the sound reduces in volume as time progresses. The Fourier cosine integral processes this fading sound, breaking it down into its oscillatory frequency components. Just as you can identify different musical notes in the fading sound, the integral reveals how each frequency contributes to the overall diminishing sound output.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Fourier Integral: It allows for representing non-periodic functions as continuous sums of sine and cosine.
-
Representation of Functions: Fourier sine and cosine integrals encapsulate different functions based on their symmetry.
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Integration Techniques: The examples illustrate techniques to calculate the Fourier integrals of piecewise continuous functions.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example of calculating the Fourier sine integral for a step function: f(x) = 1 for 0 < x < a.
-
Example of finding the Fourier cosine integral for f(x) = e^{-ax}, showing how exponential decay is analyzed in cosine form.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Sine for odd, cosine for even, make your integrals feel like heaven!
📖 Fascinating Stories
-
Imagine a wave traveling on a string - odd waves dance to sine’s tune, while even waves thrive in the cozy embrace of cosine.
🧠 Other Memory Gems
-
Remember: Sine = Odd, Cosine = Even. S = O, C = E.
🎯 Super Acronyms
FSC
- Fourier Sine for odd
- Fourier Cosine for even.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Fourier Sine Integral
Definition:
An integral representation that expresses a function using sine terms, often used for odd functions.
-
Term: Fourier Cosine Integral
Definition:
An integral representation that expresses a function using cosine terms, typically used for even functions.