A **Fourier series** is a way to represent a periodic function as a sum of simple sine and cosine waves. This technique is widely used in fields like signal processing, physics, and engineering, allowing complex waveforms to be broken down into basic components.

### Definition:
If \( f(x) \) is a periodic function with period \( 2\pi \), the Fourier series expansion of \( f(x) \) can be written as:

\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(nx) + b_n \sin(nx) \right]
\]

Here:
- \( a_0 \) is the average value of the function over one period.
- \( a_n \) and \( b_n \) are the Fourier coefficients, representing the amplitudes of the cosine and sine components for each frequency.

### Fourier Coefficients:
The coefficients are calculated as follows:

1. \( a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx \)
2. \( a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \)
3. \( b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \)

These coefficients describe how much of each sine and cosine wave is needed to reconstruct the original function.

### Example:
Consider the function \( f(x) = x \) defined on the interval \( [-\pi, \pi] \). Its Fourier series is:

\[
f(x) = 2 \sum_{n=1,3,5,\ldots}^\infty \frac{(-1)^{(n-1)/2}}{n} \sin(nx)
\]

This series consists only of sine terms because the function \( f(x) = x \) is odd.

### Applications:
1. **Signal Processing:** Used to analyze periodic signals in terms of frequency components.
2. **Vibration Analysis:** Decomposition of complex vibrations into fundamental frequencies.
3. **Heat Transfer:** Used in solving the heat equation and other partial differential equations.
4. **Quantum Mechanics:** Basis for analyzing wave functions and probability distributions.

Would you like more details on how to compute these coefficients, or another example?

### Related Questions:
1. What is the physical significance of Fourier coefficients?
2. How do you compute the Fourier series of a non-periodic function?
3. What is the difference between Fourier series and Fourier transform?
4. How is Fourier series used in signal processing?
5. Can Fourier series be used for functions with discontinuities?

### Tip:
When computing Fourier series for complex functions, symmetries like odd/even functions can simplify calculations, often eliminating either sine or cosine terms.

Explore the concept of Fourier series, a powerful tool to represent periodic functions using sine and cosine waves. This guide provides key formulas, definitions, and applications in fields like signal processing, vibration analysis, and heat transfer. Suitable for undergraduate students in engineering, physics, and applied mathematics.

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