Question

(a) Find the Fourier series of period 2 for \(f(x)=(x-1)^{2}\) on (0,2) (b) Use your result in (a) to evaluate \(\sum 1 / n^{4}\).

Short answer

The Fourier series of 2 for f(x)=(x-1)^{2} ob\tiain in Step 3. The summation of 1/n'{4}' evaluated through the result 2L in coefficient.

Step-by-step solution

Calculate the Fourier Coefficients

For a function of period 2, the Fourier series is given by: a[0] = (1/ P) * integral_{-L}^{L} f(x) dx and (24 *).

Substitute the values into the Fourier Series

Substitute the calculated Fourier coefficients into the Fourier series formula to obtain the full representation of the Fourier series of f(x).

Key concepts

Fourier coefficients

Fourier coefficients are essential components when working with Fourier series. They provide the weights for the different sine and cosine terms in the series. For a given function with period 2, we calculate the coefficients using integrals. The coefficients are divided into three types:

• The constant term, denoted as \(a_0\).
• The cosine coefficients, denoted as \(a_n\).
• The sine coefficients, denoted as \(b_n\).

The constant term \(a_0\) represents the average value of the function over one period. It's calculated by integrating the function over the period and dividing by the period length:
\[a_0 = \frac{1}{P} \int_{-L}^{L} f(x) \ dx\]
The cosine and sine coefficients, \(a_n\) and \(b_n\), are found similarly but involve multiplying the function by cosine or sine before integrating. This determines how much of each frequency component is present in the function:
\[a_n = \frac{2}{P} \int_{-L}^{L} f(x) \cos(\frac{n \pi x}{L}) \ dx\]
\[b_n = \frac{2}{P} \int_{-L}^{L} f(x) \sin(\frac{n \pi x}{L}) \ dx\]By calculating these coefficients, we capture the fundamental frequencies present in the function.

Periodic functions

A function is considered periodic if it repeats its values in regular intervals or periods. In mathematical terms, a function \(f(x)\) is periodic with period \(T\) if for some positive number \(T\), the following condition holds true for all values of \(x\):
\[f(x + T) = f(x)\]
Many natural phenomena and signals are periodic, making them important in fields like signal processing, acoustics, and electrical engineering.
When analyzing periodic functions, Fourier series become a powerful tool. They allow us to decompose a complex periodic function into simpler sine and cosine waves. This decomposition makes it easier to study and analyze the function's behavior, identify its fundamental frequencies, and understand its overall structure.

Series representation

The idea behind the series representation, specifically the Fourier series, is to express a periodic function as an infinite sum of sines and cosines. Mathematically, a Fourier series for a function \(f(x)\) of period \(2L\) is written as:

\[f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos \left( \frac{n \pi x}{L} \right) + b_n \sin \left( \frac{n \pi x}{L} \right) \right)\]
This representation provides several benefits:

• It simplifies the analysis of periodic functions.
• It allows for efficient computation and reconstruction of signals.
• It reveals the function’s frequency components, which are essential in various applications.

In the given exercise, by finding the Fourier series of \(f(x)=(x-1)^{2}\), we express this quadratic function as a sum of sines and cosines. Once we have the series, we can use it to find interesting mathematical results, like evaluating infinite sums involving powers, such as \( \sum \frac{1}{n^4} \), showcasing the power of Fourier analysis.