Math Problem Statement

Find the Fourier sine series on [0, π] of the function f(x) = {0, 0 ≤ x < π/2; π - x, π/2 ≤ x ≤ π}

Solution

To find the Fourier sine series for the given piecewise function f(x)f(x) on the interval [0,π][0, \pi], we need to follow these steps:

1. Fourier Sine Series General Formula

For a function f(x)f(x) defined on [0,π][0, \pi], the Fourier sine series is given by:

f(x)n=1bnsin(nπxπ)f(x) \sim \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{\pi}\right) where bn=2π0πf(x)sin(nπxπ)dxb_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin\left(\frac{n\pi x}{\pi}\right) dx

In this case, the function is defined as:

\begin{cases} 0 & 0 \leq x < \frac{\pi}{2} \\ \pi - x & \frac{\pi}{2} \leq x \leq \pi \end{cases}$$ ### 2. Calculate $$b_n$$ We will break the integral into two parts based on the piecewise function. $$b_n = \frac{2}{\pi} \left( \int_0^{\frac{\pi}{2}} 0 \cdot \sin\left(\frac{n\pi x}{\pi}\right) dx + \int_{\frac{\pi}{2}}^{\pi} (\pi - x) \sin\left(\frac{n\pi x}{\pi}\right) dx \right)$$ The first integral evaluates to zero because the function is zero over the interval $$[0, \frac{\pi}{2}]$$. Thus, we are left with: $$b_n = \frac{2}{\pi} \int_{\frac{\pi}{2}}^{\pi} (\pi - x) \sin\left(n x\right) dx$$ ### 3. Solve the Integral We now need to compute this integral. Let me calculate this step for you. The integral evaluates to the following result for $$b_n$$: $$b_n = \frac{2}{\pi} \left( \frac{\pi \cos\left(\frac{n\pi}{2}\right)}{2n} + \frac{\sin\left(\frac{n\pi}{2}\right)}{n^2} - \frac{\sin(n\pi)}{n^2} \right)$$ Since $$\sin(n\pi) = 0$$ for all integers $$n$$, this term vanishes. The final expression for $$b_n$$ simplifies to: $$b_n = \frac{2}{\pi} \left( \frac{\pi \cos\left(\frac{n\pi}{2}\right)}{2n} + \frac{\sin\left(\frac{n\pi}{2}\right)}{n^2} \right)$$ This gives the coefficients for the Fourier sine series of the given piecewise function. Would you like further details on this solution or clarification on any steps? Here are some related questions: 1. How do Fourier series coefficients change for different boundary conditions? 2. Can we apply this method to other piecewise functions? 3. What is the physical interpretation of the Fourier sine series? 4. How does the series converge to the original function? 5. What are the orthogonality conditions for sine functions in Fourier series? **Tip**: When solving Fourier series problems, symmetry in the function can often simplify calculations, so always check for even or odd symmetries first!

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Math Problem Analysis

Mathematical Concepts

Fourier Series
Piecewise Functions
Trigonometry

Formulas

Fourier sine series: f(x) ~ Σ(b_n sin(nπx/π))
Fourier coefficient: b_n = (2/π) ∫[0,π] f(x) sin(nπx/π) dx

Theorems

Fourier series expansion for periodic functions

Suitable Grade Level

University (Calculus/Advanced Mathematics)