To find the Fourier series for $f(x) = x$ in the interval $(-\pi, \pi)$, follow these steps:

General Fourier Series Formula:

$f(x) = a_0 + \sum_{n=1}^\infty \left( a_n \cos(nx) + b_n \sin(nx) \right)$ where:

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

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Step 1: Compute $a_0$

$a_0 = \frac{1}{2\pi} \int_{-\pi}^\pi x \, dx$ Since $x$ is an odd function and the limits are symmetric about zero: $a_0 = 0$

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Step 2: Compute $a_n$

$a_n = \frac{1}{\pi} \int_{-\pi}^\pi x \cos(nx) \, dx$ Because $x \cos(nx)$ is an odd function (product of odd $x$ and even $\cos(nx)$), the integral over symmetric limits becomes: $a_n = 0$

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Step 3: Compute $b_n$

$b_n = \frac{1}{\pi} \int_{-\pi}^\pi x \sin(nx) \, dx$ Since $x \sin(nx)$ is an even function (product of odd $x$ and odd $\sin(nx)$), evaluate the integral: $b_n = \frac{1}{\pi} \int_{-\pi}^\pi x \sin(nx) \, dx = \frac{2}{\pi} \int_{0}^\pi x \sin(nx) \, dx$ Use integration by parts:

• Let $u = x$, $dv = \sin(nx) \, dx$
• Then $du = dx$, $v = -\frac{1}{n} \cos(nx)$

$\int x \sin(nx) \, dx = -\frac{x}{n} \cos(nx) + \frac{1}{n} \int \cos(nx) \, dx$ $= -\frac{x}{n} \cos(nx) + \frac{1}{n^2} \sin(nx)$ Evaluate from $0$ to $\pi$: $\int_0^\pi x \sin(nx) \, dx = \left[ -\frac{x}{n} \cos(nx) + \frac{1}{n^2} \sin(nx) \right]_0^\pi$ At $x = \pi$: $-\frac{\pi}{n} \cos(n\pi) + \frac{1}{n^2} \sin(n\pi)$ At $x = 0$: $0$ Since $\sin(n\pi) = 0$ and $\cos(n\pi) = (-1)^n$: $\int_0^\pi x \sin(nx) \, dx = -\frac{\pi}{n} (-1)^n = \frac{\pi}{n} (-1)^{n+1}$ Thus: $b_n = \frac{2}{\pi} \cdot \frac{\pi}{n} (-1)^{n+1} = \frac{2}{n} (-1)^{n+1}$

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Fourier Series:

Substituting into the Fourier series formula: $f(x) = \sum_{n=1}^\infty \frac{2}{n} (-1)^{n+1} \sin(nx)$ Thus, the Fourier series for $f(x) = x$ in $(-\pi, \pi)$ is: $f(x) = 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx)$

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Would you like a detailed explanation of the derivation steps or any clarifications? 😊

5 Related Questions:

1. How do even and odd functions simplify Fourier series computations?
2. Why does $a_n$ vanish for odd functions?
3. What are practical applications of Fourier series in signal processing?
4. How does symmetry in $f(x)$ influence the Fourier coefficients?
5. Can this Fourier series be used to approximate $f(x) = x$ for all $x$?

Tip:

Remember that Fourier series coefficients reveal much about the function's symmetry properties—odd functions only have sine terms, and even functions only have cosine terms.