The Fourier series is a powerful mathematical tool used to represent periodic functions as a sum of sines and cosines. Below is a step-by-step breakdown to help you understand its core concepts, supplemented with examples from your uploaded material.

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Core Concepts of Fourier Series

1. Purpose of Fourier Series:

   • It decomposes any periodic function $f(x)$ into a sum of simpler sinusoidal functions (sines and cosines).

2. General Form of a Fourier Series: For a function $f(x)$ defined on the interval $[-π, π]$: $f(x) = a_0 + \sum_{n=1}^\infty \left( a_n \cos(nx) + b_n \sin(nx) \right)$ where:

   • $a_0$ is the average value of the function.
   • $a_n$ and $b_n$ are coefficients that determine the amplitude of the cosine and sine terms.

3. Fourier Coefficients:

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

4. Steps to Find a Fourier Series:

   • Step 1: Identify the function $f(x)$ and its domain.
   • Step 2: Compute $a_0$, $a_n$, and $b_n$ using the formulas above.
   • Step 3: Substitute these coefficients into the Fourier series formula.

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Example: Fourier Series of a Square Wave

Consider the function $f(x)$ given by:

\begin{cases} 0, & -\pi < x < 0, \\ 1, & 0 \leq x < \pi.$$ 1. **Compute $$a_0$$:** $$a_0 = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) dx = \frac{1}{2\pi} \left( \int_{-\pi}^0 0 dx + \int_0^\pi 1 dx \right) = \frac{\pi}{2\pi} = \frac{1}{2}.$$ 2. **Compute $$a_n$$:** $$a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) dx.$$ The integral simplifies to 0 for all $$n$$ because $$\cos(nx)$$ oscillates symmetrically. 3. **Compute $$b_n$$:** $$b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) dx.$$ Substituting $$f(x)$$: $$b_n = \frac{1}{\pi} \int_0^\pi \sin(nx) dx = \frac{1}{n\pi} \left[ -\cos(nx) \right]_0^\pi.$$ Using $$\cos(n\pi) = (-1)^n$$: $$b_n = \frac{2}{n\pi} \quad \text{for odd } n, \quad b_n = 0 \quad \text{for even } n.$$ 4. **Final Fourier Series:** $$f(x) = \frac{1}{2} + \sum_{n=1, n \text{ odd}}^\infty \frac{2}{n\pi} \sin(nx).$$ --- ### **Understanding Through Partial Sums** The Fourier series is often approximated using partial sums $$S_N(x)$$: $$S_N(x) = a_0 + \sum_{n=1}^N \left( a_n \cos(nx) + b_n \sin(nx) \right).$$ For the square wave example: $$S_1(x) = \frac{1}{2} + \frac{2}{\pi} \sin(x),$$ $$S_3(x) = \frac{1}{2} + \frac{2}{\pi} \sin(x) + \frac{2}{3\pi} \sin(3x).$$ --- Would you like me to explain another example from your material or dive deeper into applications like Gibbs phenomenon? Let me know! --- ### **Further Questions** 1. What are the applications of Fourier series in real life? 2. How does the symmetry of a function affect its Fourier coefficients? 3. What happens when $$f(x)$$ is an even or odd function? 4. How can we derive the Fourier series for non-standard intervals, e.g., $$[0, L]$$? 5. What is the Gibbs phenomenon, and why does it occur? --- ### **Tip:** Always check if the function $$f(x)$$ is even, odd, or neither. For even functions, $$b_n = 0$$; for odd functions, $$a_n = 0$$. This can save a lot of computation!