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Magazine Chapter 4: Problem 21
Let \(f(x)\) be defined for \(-\pi
Short Answer
Expert verified
The Fourier series converges to \(f(x) = \cos\left(\frac{3}{2}x \right)\); at \(x=n\cdot\frac{\pi}{2}\), the sum is \(\cos \left(\frac{3n\pi}{4} \right)\).
Step by step solution
01
Understand the Problem
We have a function defined as a periodic extension of a cosine function: \( f(x) = \cos\left(\frac{3}{2}x \right) \) for \(-\pi < x < \pi\) and \( f(x) = f(x + 2\pi) \) otherwise. We need to find its Fourier series and check convergence for all \(x\). Finally, we'll evaluate the series at special points \(x = n \cdot \frac{\pi}{2}\).
02
Identify the Period
Since \( f(x) = f(x + 2\pi) \), the function is periodic with period \(2\pi\). Therefore, the Fourier series will also have period \(2\pi\).
03
Write the Fourier Series Form
The Fourier series for a function \(f(x)\) with period \(2\pi\) is given by: \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \] where the coefficients \(a_0\), \(a_n\), and \(b_n\) are determined by integrating over one period.
04
Calculate the Coefficient \(a_0\)
The coefficient \(a_0\) is given by: \[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} \cos\left(\frac{3}{2}x \right) \, dx \].Due to symmetry, the integral of a cosine over its full period yields zero: \[ a_0 = 0 \].
05
Calculate the Coefficients \(a_n\)
The coefficients \(a_n\) are determined by: \[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} \cos \left(\frac{3}{2} x\right) \cos(nx) \, dx \]. This integral is non-zero only for \( n = \frac{3}{2} \). However, \(n\) must be an integer, and there is no positive integer that satisfies this condition in general. Therefore, \[ a_n = 0 \].
06
Calculate the Coefficients \(b_n\)
The coefficients \(b_n\) are given by: \[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} \cos \left( \frac{3}{2} x \right) \sin(nx) \, dx \]. This integral is generally non-zero when \( n = \frac{3}{2} \), which again is not an integer, leading to:\[ b_n = 0 \].
07
Conclude the Fourier Series
Given that both \(a_n\) and \(b_n\) are zero for all \(n eq \frac{3}{2}\) (which are integers), the Fourier series simplifies to only the function itself without any terms belonging to the series expansion. Thus, \[ f(x) = \cos\left(\frac{3}{2}x \right)\] on \(-\pi < x < \pi \).
08
Investigate Convergence
For a periodic function such as a cosine, the Fourier series converges to \(f(x)\) itself if \(f\) is continuous. Since \(f(x) = \cos\left(\frac{3}{2}x \right)\) is continuous and periodic with period \(2\pi\), the series converges to \(f(x)\) everywhere.
09
Evaluate at Special Points
To find the sum at \(x = n\cdot\frac{\pi}{2}\): Since the Fourier series represents the function itself, for any integer \(n\), \[ f\left(n \cdot \frac{\pi}{2} \right) = \cos \left(\frac{3n\pi}{4} \right) \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence of Series
A series is said to converge if the partial sums of the series approach a certain value as more and more terms are added. For a Fourier series representing a periodic function, convergence implies that as we add more terms in the series, the result becomes increasingly close to the function it represents. One important condition for the convergence of a Fourier series is that the function it represents is continuous over its domain.
In the case of the periodic function defined as \( f(x) = \cos \left(\frac{3}{2}x \right) \), the function is continuous between \(-\pi\) and \(\pi\). This continuity ensures that the Fourier series will converge to the function itself across its domain. The convergence of a Fourier series to a piecewise smooth and continuous function is guaranteed by Dirichlet's conditions, which assure us that the series will appropriately represent the function at every point.
In the case of the periodic function defined as \( f(x) = \cos \left(\frac{3}{2}x \right) \), the function is continuous between \(-\pi\) and \(\pi\). This continuity ensures that the Fourier series will converge to the function itself across its domain. The convergence of a Fourier series to a piecewise smooth and continuous function is guaranteed by Dirichlet's conditions, which assure us that the series will appropriately represent the function at every point.
- Continuity helps ensure the series converges.
- Fourier series converges to the actual function where it is continuous.
- Discontinuities in the function may result in non-convergence at specific points.
Periodic Functions
Periodic functions are functions that repeat their values over regular intervals. A function \(f(x)\) is said to be periodic with period \(T\) if \(f(x+T) = f(x)\) for all values of \(x\). The function defined in the exercise, \(f(x) = \cos \left(\frac{3}{2}x \right)\), continues these values for all \(x\) with the rule \(f(x) = f(x + 2\pi)\). Thus, it is a periodic function with the period \(2\pi\).
Periodic functions are fundamental in the study of Fourier series because such functions can be expressed as a sum of sines and cosines, each of which has properties of periodicity. This decomposition into simpler oscillatory components allows us to analyze complex periodic phenomena through their Fourier series.
Periodic functions are fundamental in the study of Fourier series because such functions can be expressed as a sum of sines and cosines, each of which has properties of periodicity. This decomposition into simpler oscillatory components allows us to analyze complex periodic phenomena through their Fourier series.
- Periodic functions repeat values over consistent intervals.
- Understanding periodicity helps in identifying the range over which the function essentially "resets".
- Fourier series are a natural tool for decomposing periodic functions.
Cosine Function
The cosine function \(\cos(x)\) is a fundamental trigonometric function exhibiting wave-like periodic behavior. The function completes one full cycle every \(2\pi\) units of \(x\). Within each cycle, \(\cos(x)\) starts from 1, decreases to -1, and returns to 1. It is even-function symmetry, which means \(\cos(-x) = \cos(x)\), making it particularly useful for representing symmetrical waveforms.
In the problem, the function is modified to \(\cos \left(\frac{3}{2}x \right)\), which changes the frequency of oscillation and affects how many cycles occur within a given interval. With \(\frac{3}{2}\) as the frequency multiplier, the function oscillates more frequently compared to a standard cosine. This is critical to forming a Fourier series representation, as our component functions must align perfectly within the defined interval.
Cosine functions are crucial in Fourier series because:
In the problem, the function is modified to \(\cos \left(\frac{3}{2}x \right)\), which changes the frequency of oscillation and affects how many cycles occur within a given interval. With \(\frac{3}{2}\) as the frequency multiplier, the function oscillates more frequently compared to a standard cosine. This is critical to forming a Fourier series representation, as our component functions must align perfectly within the defined interval.
Cosine functions are crucial in Fourier series because:
- They maintain even symmetry, simplifying calculations.
- Alterations in frequency multiply the number of times the function crosses its mean value within one period.
- They contribute to expressing and analyzing periodic functions efficiently within a Fourier series.
