Expand the following functions in Legendre series. $$f(x)=\left\\{\begin{array}{lr} 0, & -1

Legendre polynomials are a type of orthogonal polynomials that are solutions to the Legendre differential equation. They are denoted by \( P_n(x) \), where \( n \) indicates the polynomial's degree. These polynomials are defined over the interval \([-1, 1]\) and have unique properties that make them very useful in approximating functions.

One of the most important properties of Legendre polynomials is that they are orthogonal with respect to the inner product:
\[ \int_{-1}^{1} P_m(x) P_n(x) \, dx = 0, \, \text{if} \, m \e n \]\
This orthogonality means we can decompose functions into sums of these polynomials, which we call a Legendre series expansion. Here are the first few Legendre polynomials:

• \( P_0(x) = 1 \)
• \( P_1(x) = x \)
• \( P_2(x) = \frac{1}{2}(3x^2 - 1) \)

Piecewise functions are functions defined by different expressions in different intervals of the domain. For example, in the given exercise:
\[ f(x)=\begin{cases} \, \ \0, & -1 < x < 0 \ \ x, & \, 0 < x < 1 \ \ \end{cases} \]

This function is defined as 0 for \(-1 < x < 0\) and as \( x \) for \( 0 < x < 1 \). This means the function takes different values depending on the interval of \( x \). Working with piecewise functions can be tricky because you need to handle each interval separately during integration or differentiation.

When expanding piecewise functions in a series, splitting the integral based on the defined intervals is crucial. This ensures that each segment of the function contributes correctly to the expansion coefficients.

Orthogonal functions are functions that satisfy an orthogonality relation, which means their inner product over a specific interval is zero unless they are identical. Mathematically, two functions \( f(x) \) and \( g(x) \) are orthogonal if:
\[ \int_{a}^{b} f(x) g(x) \, dx = 0 \]
Orthogonal functions play a crucial role in many areas of mathematics and physics, especially in solving differential equations and in decomposing functions into series.

Legendre polynomials are a classic example of orthogonal functions. Because of their orthogonality, any function defined over \([-1, 1]\) can be expressed as an infinite series of these polynomials. This decomposition simplifies the process of analyzing and working with complex functions.

Orthogonal functions ensure that each term in the series expansion impacts the summation uniquely without interfering with other terms. This property helps in simplifying and solving various mathematical problems efficiently.

The Legendre series expansion of a function involves determining coefficients that multiply each Legendre polynomial term in the series. These coefficients \( a_n \) are found using the formula:
\[ a_n = \frac{2n + 1}{2} \int_{-1}^{1} f(x) P_n(x) \, dx \]
These integrals measure how much each Legendre polynomial contributes to representing the function over its domain.

For piecewise functions, the integral is split into parts corresponding to the function's defined intervals. In the given exercise, the function is defined on two intervals. The integral will be:
\[ a_n = \frac{2n + 1}{2} \left( \int_{-1}^{0} 0 \cdot P_n(x) \, dx + \int_{0}^{1} x \cdot P_n(x) \, dx \right) \]
The first integral is zero, simplifying the calculation. Evaluating the second integral for different \(n\) values gives specific series coefficients that describe the function in terms of Legendre polynomials.

For example:

• \( a_0 = \frac{2(0) + 1}{2} \int_{0}^{1} x \, dx = \frac{1}{2} \frac{x^2}{2} \big\bigg|_{0}^{1} = \frac{1}{4} \)

Mastering the calculation of these coefficients helps in constructing an accurate Legendre series expansion.