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Magazine Chapter 4: Problem 7
Suppose \(f(t)\) is defined on \((-\pi, \pi]\) as \(t^{3} .\) Extend periodically and compute the Fourier series of \(f(t)\).
Short Answer
Expert verified
The Fourier series is \(\sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{n} \sin(nt)\).
Step by step solution
01
Understand the Problem
We are given a function \(f(t) = t^3\) defined on the interval \((-\pi, \pi]\). Our task is to extend this function periodically and to find its Fourier series.
02
Determine the Periodicity
Since the function is defined on \((-\pi, \pi]\), the period of the function is \(2\pi\). Thus, we intend to find a Fourier series that represents this periodic extension.
03
Identify Fourier Series Components
A Fourier series for a function with period \(2\pi\) is given by: \[a_0 + \sum_{n=1}^{\infty}(a_n \cos(nt) + b_n \sin(nt))\]where \(a_0\), \(a_n\), and \(b_n\) are Fourier coefficients to be determined.
04
Calculate \(a_0\) Coefficient
The zeroth coefficient \(a_0\) is calculated as: \[a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} t^3 \, dt\]Evaluate the integral: \[a_0 = \frac{1}{2\pi} \left[ \frac{t^4}{4} \right]_{-\pi}^{\pi} = \frac{1}{2\pi}\left( \frac{\pi^4}{4} + \frac{\pi^4}{4} \right) = 0\]
05
Determine \(a_n\) Coefficients
The coefficients \(a_n\) are calculated by: \[a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} t^3 \cos(nt) \, dt\]This integral is zero due to symmetry, as \(t^3\cos(nt)\) is an odd function.
06
Determine \(b_n\) Coefficients
Coefficients \(b_n\) are calculated using: \[b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} t^3 \sin(nt) \, dt\]This results in: \[b_n = \frac{2(-1)^{n+1}}{n}\] for \(n eq 0\), calculated via integration by parts.
07
Compile the Fourier Series
Since \(a_0 = 0\) and \(a_n = 0\) for all \(n\), the Fourier series comprises only sine terms: \[f(t) \sim \sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{n} \sin(nt)\] This is the Fourier series representation of \(f(t) = t^3\) over \((-\pi, \pi]\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Periodic Functions
A periodic function is a function that repeats its values at regular intervals or periods. Understanding periodic functions is crucial here because we need to extend the function \(f(t)=t^3\) periodically over the interval \((-\pi, \pi]\).This way, the pattern of the function repeats every \(2\pi\).
In simpler terms, imagine a wave that starts at \(-\pi\), peaks, troughs, and ends at \(\pi\) where it starts the same pattern again. When we extend \(f(t)=t^3\) periodically, it means we'll take this cubic function confined to \((-\pi, \pi]\) and repeat its graph beyond this interval continuously.
In simpler terms, imagine a wave that starts at \(-\pi\), peaks, troughs, and ends at \(\pi\) where it starts the same pattern again. When we extend \(f(t)=t^3\) periodically, it means we'll take this cubic function confined to \((-\pi, \pi]\) and repeat its graph beyond this interval continuously.
- Characteristic: The function values repeat after a fixed period: \(f(t + 2\pi) = f(t)\) for all \(t\).
- Application: Essential for defining functions on the whole real line via Fourier series.
Fourier Coefficients
Fourier coefficients are key components in expressing a function as a Fourier series.The challenge in decomposing our function \(f(t) = t^3\) into a series is finding these coefficients.
The coefficients \(a_0\), \(a_n\), and \(b_n\) define how much of each trigonometric component is in our function.
The coefficients \(a_0\), \(a_n\), and \(b_n\) define how much of each trigonometric component is in our function.
- \(a_0\) is the average or mean value of the function over its period. It acts as a baseline around which other variations occur.
- \(a_n\) are the coefficients of cosine terms, capturing even symmetrical components. These are zero in this problem, indicating symmetry cancellation.
- \(b_n\) are the coefficients of sine terms, capturing the function's odd symmetry. For this problem, \(b_n = \frac{2(-1)^{n+1}}{n}\).
Integration by Parts
Integration by parts is a mathematical technique used to integrate the product of two functions. This technique is crucial for finding the Fourier coefficient \(b_n\) for \(f(t) = t^3\).
The formula for integration by parts is:\[\int u \ dv = uv - \int v \ du\]where \(u\) and \(dv\) are chosen parts of the integrand.
The formula for integration by parts is:\[\int u \ dv = uv - \int v \ du\]where \(u\) and \(dv\) are chosen parts of the integrand.
- Typically, \(u\) is a function that simplifies when differentiated (e.g., polynomials like \(t^3\)).
- \(dv\) is chosen so its integral \(v\) is easy to find (e.g., \(\sin(nt)\) or \(\cos(nt)\)).
- During integration, carefully consider which function to assign to \(u\) and \(dv\) to simplify calculations.
