Math Problem Statement
7th qns answer
The Fourier series of the function f(x) defined as x - [x] - 1/2 for non-integer x and x for integer x, where [x] is the greatest integer function in the interval [-1/2, 1/2].
Solution
The question asks for the Fourier series of a given function . The function is defined as:
\begin{cases} x - \lfloor x \rfloor - \frac{1}{2}, & \text{if } x \text{ is not an integer}, \\ 0, & \text{if } x \text{ is an integer}, \end{cases}$$ where $$\lfloor x \rfloor$$ is the greatest integer function (also known as the floor function), within the interval $$\left[ -\frac{1}{2}, \frac{1}{2} \right]$$. Given the periodicity of $$f(x)$$ with $$f(x+1) = f(x)$$, this is a typical setup for finding the Fourier series of a periodic function. ### Fourier Series For a periodic function $$f(x)$$ with period $$2L$$, the Fourier series can be expressed as: $$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right),$$ if the function is odd (as $$f(x)$$ is in this case). Here, the coefficients $$b_n$$ are given by: $$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx.$$ ### For this function: - The function is defined to have period 1, so $$L = \frac{1}{2}$$. - The function is odd, so we only expect sine terms in the Fourier series expansion. Looking at the choices, the correct Fourier series representation should be: $$f(x) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\sin(n\pi x)}{n}.$$ Thus, the correct answer is the second option: $$\boxed{f(x) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\sin(n\pi x)}{n}}.$$ Would you like further details or clarification on this? Here are some related questions: 1. What is the general process for finding the Fourier coefficients for a periodic function? 2. Why are there only sine terms in the Fourier series of this function? 3. How does the symmetry of a function influence its Fourier series expansion? 4. What is the significance of the greatest integer function $$\lfloor x \rfloor$$ in this context? 5. How do the Fourier coefficients $$b_n$$ relate to the function's behavior at different points? **Tip**: Always check the symmetry of a function (odd or even) before deciding which terms (sine or cosine) will appear in its Fourier series.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Fourier Series
Periodic Functions
Greatest Integer Function
Formulas
Fourier Series: f(x) = sum from n=1 to infinity of b_n sin(nπx/L)
Fourier Coefficients: b_n = (2/L) ∫ from 0 to L of f(x) sin(nπx/L) dx
Theorems
Fourier Series Expansion for Periodic Functions
Symmetry in Fourier Series
Suitable Grade Level
Undergraduate
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