Determine the Fourier Coefficients

Oct 13, 2025 by miodeya.com 127 views

Determine the Fourier Series to Represent the Function f(x) = |x| in (-1 < x < 1)

The Fourier series is a mathematical tool used to represent periodic functions as a sum of sinusoidal functions. It is a powerful technique for analyzing and solving problems in physics, engineering, and other fields. In this article, we will determine the Fourier series to represent the function f(x) = |x| in the interval (-1 < x < 1).

What is the Fourier Series?

The Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is defined as:

f(x) = a0 + ∑[n=1 to ∞] (an cos(nx) + bn sin(nx))

where a0, an, and bn are the Fourier coefficients.

The Function f(x) = |x|

The function f(x) = |x| is a piecewise function defined as:

f(x) = x for 0 ≤ x < 1 f(x) = -x for -1 < x < 0

Determine the Fourier Coefficients

To determine the Fourier coefficients, we need to calculate the following integrals:

a0 = (1/2) ∫[-1,1] f(x) dx an = (1/π) ∫[-1,1] f(x) cos(nx) dx bn = (1/π) ∫[-1,1] f(x) sin(nx) dx

Calculate a0

To calculate a0, we need to evaluate the following integral:

a0 = (1/2) ∫[-1,1] f(x) dx = (1/2) ∫[-1,0] -x dx + (1/2) ∫[0,1] x dx = (1/2) [-x^2/2] from -1 to 0 + (1/2) [x^2/2] from 0 to 1 = (1/2) [-1/2 + 1/2] + (1/2) [1/2 - 0] = 1/2

Calculate an

To calculate an, we need to evaluate the following integral:

an = (1/π) ∫[-1,1] f(x) cos(nx) dx = (1/π) ∫[-1,0] -x cos(nx) dx + (1/π) ∫[0,1] x cos(nx) dx

Using integration by parts, we get:

an = (1/π) [-x sin(nx)/n] from -1 to 0 + (1/π) [x sin(nx)/n] from 0 to 1

(1/π) ∫[-1,0] sin(nx) dx - (1/π) ∫[0,1] sin(nx) dx = (1/π) [-x sin(nx)/n] from -1 to 0 + (1/π) [x sin(nx)/n] from 0 to 1
(1/π) [-cos(nx)/n] from -1 to 0 - (1/π) [cos(nx)/n] from 0 to 1

Evaluating the limits, we get:

an = (1/π) [1/n] - (1/π) [1/n] = 0

Calculate bn

To calculate bn, we need to evaluate the following integral:

bn = (1/π) ∫[-1,1] f(x) sin(nx) dx = (1/π) ∫[-1,0] -x sin(nx) dx + (1/π) ∫[0,1] x sin(nx) dx

Using integration by parts, we get:

bn = (1/π) [-x cos(nx)/n] from -1 to 0 + (1/π) [x cos(nx)/n] from 0 to 1

(1/π) ∫[-1,0] cos(nx) dx - (1/π) ∫[0,1] cos(nx) dx = (1/π) [-x cos(nx)/n] from -1 to 0 + (1/π) [x cos(nx)/n] from 0 to 1
(1/π) [sin(nx)/n] from -1 to 0 - (1/π) [sin(nx)/n] from 0 to 1

Evaluating the limits, we get:

bn = (1/π) [1/n] - (1/π) [-1/n] = 2/nπ

Determine the Fourier Series

Now that we have calculated the Fourier coefficients, we can determine the Fourier series:

f(x) = a0 + ∑[n=1 to ∞] (an cos(nx) + bn sin(nx)) = 1/2 + ∑[n=1 to ∞] (0 cos(nx) + 2/nπ sin(nx))

Simplifying the expression, we get:

f(x) = 1/2 + ∑[n=1 to ∞] (2/nπ sin(nx))

Conclusion

In this article, we determined the Fourier series to represent the function f(x) = |x| in the interval (-1 < x < 1). We calculated the Fourier coefficients a0, an, and bn using the given integrals. The resulting Fourier series is:

f(x) = 1/2 + ∑[n=1 to ∞] (2/nπ sin(nx))

Q: What is the Fourier series for f(x) = |x| in the interval (-1 < x < 1)?

A: The Fourier series for f(x) = |x| in the interval (-1 < x < 1) is:

f(x) = 1/2 + ∑[n=1 to ∞] (2/nπ sin(nx))

Q: How do I calculate the Fourier coefficients a0, an, and bn?

A: To calculate the Fourier coefficients a0, an, and bn, you need to evaluate the following integrals:

a0 = (1/2) ∫[-1,1] f(x) dx an = (1/π) ∫[-1,1] f(x) cos(nx) dx bn = (1/π) ∫[-1,1] f(x) sin(nx) dx

Q: What is the significance of the Fourier series?

A: The Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is a powerful technique for analyzing and solving problems in physics, engineering, and other fields.

Q: Can I use the Fourier series to represent any function?

A: No, the Fourier series can only be used to represent periodic functions. If the function is not periodic, you may need to use other mathematical techniques to represent it.

Q: How do I determine the number of terms to include in the Fourier series?

A: The number of terms to include in the Fourier series depends on the desired level of accuracy. In general, you can include more terms to achieve higher accuracy, but this may also increase the computational complexity.

Q: Can I use the Fourier series to solve real-world problems?

A: Yes, the Fourier series can be used to solve a wide range of real-world problems, including signal processing, image analysis, and vibration analysis.

Q: What are some common applications of the Fourier series?

A: Some common applications of the Fourier series include:

Signal processing: The Fourier series can be used to analyze and process signals in various fields, such as audio and image processing.
Image analysis: The Fourier series can be used to analyze and process images in various fields, such as medical imaging and computer vision.
Vibration analysis: The Fourier series can be used to analyze and process vibrations in various fields, such as mechanical engineering and aerospace engineering.

Q: How do I implement the Fourier series in a programming language?

A: The implementation of the Fourier series in a programming language depends on the specific language and the desired level of accuracy. In general, you can use libraries such as NumPy and SciPy in Python to implement the Fourier series.

Q: What are some common pitfalls to avoid when using the Fourier series?

A: Some common pitfalls to avoid when using the Fourier series include:

Using the Fourier series to represent non-periodic functions.
Including too few terms in the Fourier series, which can lead to inaccurate results.
Using the Fourier series to solve problems that are not well-suited for this technique.

Q: Can I use the Fourier series to represent complex functions?

A: Yes, the Fourier series can be used to represent complex functions. However, this may require the use of complex analysis and the extension of the Fourier series to the complex plane.

Q: How do I extend the Fourier series to the complex plane?

A: To extend the Fourier series to the complex plane, you need to use complex analysis and the extension of the Fourier series to the complex plane. This may involve the use of complex integrals and the application of the Cauchy-Riemann equations.