Question

Expand the following in a Fourier-Legendre series for \(x \in(-1,1)\). a. \(f(x)=x^{2}\). b. \(f(x)=5 x^{4}+2 x^{3}-x+3\). c. \(f(x)=\left\\{\begin{array}{cc}-1, & -1

Short answer

The solution will consist of four Fourier-Legendre series, one for each given function.

Step-by-step solution

Write the general expression for Fourier-Legendre series

The general expression for a Fourier-Legendre series of a function \(f(x)\) is \(\displaystyle f(x)=\sum a_{n} P_{n}(x)\), where \(P_n(x)\) are the Legendre polynomials.

Coefficient Calculation

To find the coefficient \(a_n\), use the formula \(\displaystyle a_{n}=\frac{2 n+1}{2} \int_{-1}^{1} f(x) P_{n}(x) d x\). Implement this formula for each given function \(f(x)\) and calculate the coefficients.

Construct the Fourier-Legendre Series For First Function

For the first function \(f(x)=x^{2}\), calculate the coefficients and substitute them into the Fourier series.

Construct the Fourier-Legendre Series For Second Function

For the second function \(f(x)=5 x^{4}+2 x^{3}-x+3\), calculate the coefficients and substitute them into the Fourier series.

Construct the Fourier-Legendre Series For Third Function

For the third function, which is defined piecewise as -1 for \(-1<x<0\) and 1 for \(0<x<1\), calculate the coefficients and substitute them into the Fourier series.

Construct the Fourier-Legendre Series For Fourth Function

For the fourth function, which is defined piecewise as \(x\) for \(-1<x<0\) and 0 for \(0<x<1\), calculate the coefficients and substitute them into the Fourier series.

Key concepts

Legendre Polynomials

Understanding Legendre polynomials is critical for expanding functions into Fourier-Legendre series. These polynomials form a set of orthogonal polynomials that are solutions to the Legendre differential equation. A noteworthy property of Legendre polynomials is that they are normalized to be equal to one at x=1, which simplifies the integration process when calculating coefficients.

For instance, P_0(x) = 1, P_1(x) = x, P_2(x) = (3x^2-1)/2, and so on. Each successive polynomial has a higher degree and contains only even or odd powers of x, mirroring the polynomial's parity. These polynomials are also useful beyond Fourier series - in physics and engineering for problems involving spherical coordinates.

Coefficient Calculation in Fourier Series

The coefficients in a Fourier series capture the essence of the original function. For a Fourier-Legendre series, the coefficients a_n determine how much of each Legendre polynomial P_n(x) is needed to reconstruct the given function. Calculating coefficients involves integrating the product of the original function and the corresponding Legendre polynomial over the interval (-1,1).

The formula a_n = (2n+1)/2 * ∫(f(x) * P_n(x) dx) from -1 to 1 is used, where n indicates the degree of the Legendre polynomial. When working through these calculations, it is crucial to remember that due to their orthogonality, any integral involving non-matching polynomial degrees will yield zero - greatly simplifying the process.

Orthogonal Polynomial Expansion

Orthogonal polynomials are a cornerstone of Fourier series expansions. Their orthogonality means that when two different polynomials of the set are multiplied together and integrated over a certain interval, the result is zero. This property is what makes them exceptionally valuable for series expansions like the Fourier-Legendre series.

Orthogonality expedites the process of computing coefficients because only terms with matching polynomial degrees will have a non-zero contribution. Thus, it acts as a 'sifting' property that isolates the contribution of each term in the expansion, allowing for a straightforward reconstruction of the original function.

Piecewise Function Representation

Piecewise functions are functions that have different expressions for different intervals of the variable's domain. These functions can sometimes present challenges in mathematical expansions, but they are nicely handled by the Fourier-Legendre series, thanks to the series' ability to approximate functions over a specified interval.

In a Fourier-Legendre series expansion, determining the coefficients for a piecewise function requires separate integrations over each interval defined by the piecewise function. However, due to the orthogonality of Legendre polynomials, each piece contributes independently to the coefficients, which is clearly reflected when integrating the function over the different intervals.