To find the Fourier series of $f(x) = x$ on $(0, \pi)$, assuming it’s extended as an even function over the interval $(-\pi, \pi)$, we’ll derive both the real (sine-cosine) and complex Fourier series representations.

1. Defining the Problem

Since $f(x) = x$ is defined as even over $(-\pi, \pi)$, its Fourier series will only contain cosine terms (since sine terms represent the odd part of a function).

Function on $(-\pi, \pi)$:

\begin{cases} x & \text{if } 0 \leq x \leq \pi \\ -x & \text{if } -\pi \leq x < 0 \end{cases}$$ ### 2. Real Fourier Series (Sine-Cosine Form) The real Fourier series of $$f(x)$$ on $$(-\pi, \pi)$$ is given by: $$f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(nx)$$ where $$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx$$ and $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx$$ #### Step 1: Calculating $$a_0$$ Since $$f(x)$$ is odd on $$(-\pi, \pi)$$ (it’s $$f(x) = x$$ on $$(0, \pi)$$ and $$f(x) = -x$$ on $$(-\pi, 0)$$), $$a_0 = 0$$. #### Step 2: Calculating $$a_n$$ Now, let’s compute $$a_n$$: $$a_n = \frac{1}{\pi} \int_{0}^{\pi} x \cos(nx) \, dx$$ Using integration by parts, we find: $$a_n = \frac{(-1)^{n+1} 2}{n}$$ So, the real Fourier series for $$f(x) = x$$ on $$(0, \pi)$$ is: $$f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} 2}{n} \cos(nx)$$ ### 3. Complex Fourier Series The complex Fourier series representation of $$f(x)$$ is given by: $$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}$$ where $$c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx$$ #### Computing $$c_n$$: After similar integration, we find: $$c_n = \frac{i(-1)^n}{n}$$ ### Summary 1. **Real Fourier Series**: $$f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} 2}{n} \cos(nx)$$ 2. **Complex Fourier Series**: $$f(x) = \sum_{n=-\infty}^{\infty} \frac{i(-1)^n}{n} e^{inx}$$ Would you like to see further steps in the integration process or have other questions? --- ### Related Questions 1. How would the Fourier series change if $$f(x) = x$$ was defined on $$(-L, L)$$ with a different interval length? 2. Can Fourier series be used to approximate non-periodic functions, and if so, how? 3. What are the differences between Fourier series and Fourier transforms in analyzing functions? 4. How does the Fourier series converge to a function, and what are the conditions for convergence? 5. How would the Fourier coefficients change if $$f(x) = |x|$$ on $$(-\pi, \pi)$$? --- ### Tip In Fourier series, functions extended as odd functions produce sine terms, while even extensions produce cosine terms, simplifying the series based on symmetry.