Question

Find the mixed Fourier cosine series. $$ f(x)=x ; \quad[0,1] $$

Short answer

The mixed Fourier cosine series for the function $$f(x) = x$$ in the interval $$[0,1]$$ is given by: $$ f(x) \approx \frac{1}{2} + \sum_{n=1}^{\infty}\frac{2}{n^2\pi^2}(\cos(n\pi) - 1) \cos(n\pi x) $$

Step-by-step solution

Compute the Fourier coefficients

To find the mixed Fourier cosine series of the function $$f(x) = x$$, we first need to find the Fourier coefficients $$a_0$$ and $$a_n$$. They can be calculated using the following formulas: $$ a_0 = \frac{2}{1-0}\int_{0}^{1} f(x) dx $$ and $$ a_n = \frac{2}{1-0}\int_{0}^{1}f(x)\cos(n\pi x)dx $$ where n is a positive integer. We will now compute the coefficients.

Calculate the a_0 coefficient

To calculate the $$a_0$$ coefficient, we need to integrate \(f(x)\) from 0 to 1: $$ a_0 = 2 \int_{0}^{1} x dx $$ Integrating this expression will give us: $$ a_0 = 2 \left[\frac{1}{2}x^2\right]_0^1 = 2(0.5) = 1 $$ Thus, the coefficient $$a_0$$ equals 1.

Calculate the a_n coefficient

To calculate the $$a_n$$ coefficient, we need to integrate the product of $$f(x)$$ and $$\cos(n\pi x)$$ in the interval [0,1]: $$ a_n = 2 \int_{0}^{1} x \cos(n\pi x) dx $$ This integral is a bit more complex, so we will use integration by parts: Let $$u = x$$ and $$dv = \cos(n\pi x) dx$$. Then, we find the derivatives and integrals: $$ du = dx \\ v = \frac{1}{n\pi}\sin(n\pi x) $$ Now, we will apply the integration by parts formula: $$ \int u \, dv = uv - \int v \, du $$ Thus, $$ a_n = 2 \left[ x \cdot \frac{1}{n\pi}\sin(n\pi x) \Big|_0^1 - \int_{0}^{1} \frac{1}{n\pi}\sin(n\pi x) dx \right] $$ Evaluating the expression inside the square brackets, we have: $$ a_n = 2 \left[ \frac{1}{n\pi}\sin(n\pi) - \int_{0}^{1}\frac{1}{n\pi}\sin(n\pi x) dx \right] $$ Simplifying, $$ a_n = \frac{2}{n\pi} \left[\int_{0}^{1} \sin(n\pi x) dx \right] $$ Now integrate the remaining integral: $$ a_n = \frac{2}{n\pi}\left[ -\frac{1}{n\pi}\cos(n\pi x) \right]_0^1 $$ Evaluating and simplifying, we have: $$ a_n = \frac{2}{n^2\pi^2}(\cos(n\pi) - 1) $$

Formulate the mixed Fourier cosine series

With the Fourier coefficients $$a_0$$ and $$a_n$$ found, we can now write the mixed Fourier cosine series for the function f(x): $$ f(x) \approx \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n\pi x) $$ Substituting the calculated values of $$a_0$$ and $$a_n$$: $$ f(x) \approx \frac{1}{2} + \sum_{n=1}^{\infty}\frac{2}{n^2\pi^2}(\cos(n\pi) - 1) \cos(n\pi x) $$ This is the mixed Fourier cosine series for the function $$f(x) = x$$ in the interval $$[0,1]$$.

Key concepts

Fourier Coefficients

Fourier coefficients are crucial in decomposing functions into a series of sines and cosines, known as Fourier series. In the context of a mixed Fourier cosine series, these coefficients help represent a function like \(f(x) = x\) over an interval as a sum of cosine terms. The formulas for the coefficients are given by \(a_0\) and \(a_n\).

Determining \(a_0\) involves integrating the function over the interval \([0, 1]\) which gives \(a_0 = 2\int_{0}^{1} x \; dx\).

Evaluating this integral leads to \(a_0 = 1\), simplifying calculations by being half of the leading term in the series. Each \(a_n\) is calculated using \(a_n = 2\int_{0}^{1} x \cos(n\pi x) \; dx\), which requires integration by parts, due to the presence of a product \((x \cdot \cos(n\pi x))\). As \(n\) increases, the contribution of each term in the series affects the result exhibitively.

Integration by Parts

Integration by parts is a technique used to integrate products of functions, especially useful when dealing with integrals like \(\int x \cos(n\pi x) \, dx\). It is based on the principle of reversing the product rule for derivatives.

The formula \(\int u \, dv = uv - \int v \, du\) allows us to choose one function as \(u\) to differentiate and another as \(dv\) to integrate. Here, \(u = x\) and \(dv = \cos(n\pi x)\, dx\), leading to \(du = dx\) and \(v = \frac{1}{n\pi}\sin(n\pi x)\). The integration by parts steps let us rewrite the integral in a form that's easier to handle.

Completing this process yields \(a_n\), revealing how each calculated coefficient contributes to the Fourier series. It shows the utility of the method in transforming more complex integrals into manageable components.

Mixed Fourier Cosine Series

A mixed Fourier cosine series is a way to express a function using only cosine terms. This series is useful for even functions or functions defined on specific intervals, like \([0, 1]\) for \(f(x) = x\).

The general form is \(f(x) \approx \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n\pi x)\). Here, \(a_0\) and \(a_n\) are Fourier coefficients found from earlier calculations, specifically tailored to fit the cosine series. The presence of \(\cos(n\pi x)\) indicates how the function is captured across its interval, utilizing symmetry and periodicity.

• \(\frac{a_0}{2}\) represents the average or mean part of the function over the interval.
• The sum of all \(a_n \cos(n\pi x)\) represents fluctuations around this average due to other frequencies captured within the function.

This makes the mixed Fourier cosine series an effective tool for breaking down complex periodic functions into understandable components of cosine functions.