Question

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

Short answer

The Legendre series expansion has only even terms with coefficients calculated as \( a_n = (2n+1) \times \int_0^1 P_n(x) \, dx \).

Step-by-step solution

Understanding the Function

The function \( f(x) = \left\{ \begin{array}{rr} -1, & -1 < x < 0 \ 1, & 0 < x < 1 \end{array} \right. \) is defined piecewise. Note the intervals where the function has different constant values.

Recall the Legendre Series Expansion Formula

The Legendre series expansion of a function \( f(x) \) on the interval \( [-1, 1] \) is given by: \[ f(x) = \sum_{n=0}^\infty a_n P_n(x) \], where \( P_n(x) \) are the Legendre polynomials and the coefficients \( a_n \) are found by integrating: \[ a_n = \frac{2n+1}{2} \int_{-1}^{1} f(x) P_n(x) \, dx \].

Integrate for Coefficients \( a_n \)

Calculate the coefficients \( a_n \) by evaluating the integral for each \( n \). \( a_n = \frac{2n+1}{2} \left[ \int_{-1}^{0} (-1)P_n(x) \, dx + \int_{0}^{1} (1)P_n(x) \, dx \right] \).

Separate Integrals

Separate the integrals into two parts: \[ a_n = \frac{2n+1}{2} \left[- \int_{-1}^{0} P_n(x) \, dx + \int_{0}^{1} P_n(x) \, dx \right].\]

Simplify Using Orthogonality

Use the fact that Legendre polynomials are orthogonal over \( [-1, 1] \). This implies that the integral of \( P_n(x) \) over \( [-1, 1] \) is zero if \( n \) is odd and non-zero if \( n \) is even.

Evaluate for Even and Odd \( n \)

For odd \( n \), the symmetric property of Legendre polynomials yields zero contributions: \[ a_n = 0.\] For even \( n \), compute: \[ a_n = (2n+1) \times \int_0^1 P_n(x) \, dx.\]

Key concepts

Piecewise Functions

A piecewise function is a function that has different expressions for different intervals of its domain. In other words, the function's definition changes based on the input value. For instance, in the given problem, the function \( f(x) \) is defined as follows:
\( f(x) = \left\{ \begin{array}{rr} -1, & -1 < x < 0 \ 1, & 0 < x < 1 \ \end{array} \right. \).

Here, the function has two distinct constant values: -1 for \(-1 < x < 0\) and 1 for \(0 < x < 1\). This change in value is why it's called a piecewise function. These types of functions are quite common in real-world scenarios like signal processing and physics, where different conditions apply over different ranges.

When dealing with piecewise functions, one challenge is representing them in a unified form, such as when performing integration or series expansion. You need to account for each piece separately before combining the results to form the overall solution.

Legendre Polynomials

Legendre polynomials, \( P_n(x) \), are a set of orthogonal polynomials that arise frequently in physics and engineering, particularly in problems involving spherical coordinates. These polynomials are solutions to Legendre's differential equation and are defined over the interval \/([-1, 1])\.

A key property of Legendre polynomials is their orthogonality, which means that the integral of the product of two different Legendre polynomials over the interval \/([-1, 1])\ is zero:
\( \int_{-1}^{1} P_m(x) P_n(x) \, dx = 0 \) if \( m eq n \).

This property simplifies the process of finding coefficients in a Legendre series expansion. The general expansion for a function \( f(x) \) using Legendre polynomials is:
\( f(x) = \sum_{n=0}^\infty a_n P_n(x) \),
where the coefficients \( a_n \) are given by:
\( a_n = \frac{2n+1}{2} \int_{-1}^{1} f(x) P_n(x) \, dx \).
These coefficients indicate how much of each polynomial contributes to the function.

Orthogonality

Orthogonality is a crucial concept in mathematics, particularly in the context of functions and polynomials. It refers to the idea that two functions, such as Legendre polynomials, are orthogonal if their inner product over a specific interval is zero. For Legendre polynomials \( P_m(x) \) and \( P_n(x) \), orthogonality means:
\( \int_{-1}^{1} P_m(x) P_n(x) \, dx = 0 \) if \( m eq n \).

This property greatly simplifies the computation of coefficients in series expansions. In the case of a Legendre series, orthogonality helps to isolate each coefficient \( a_n \)
in the expansion:
\( a_n = \frac{2n+1}{2} \int_{-1}^{1} f(x) P_n(x) \, dx \).

Thanks to orthogonality, when you integrate the product of the function and the polynomial, all terms involving other polynomials vanish, leaving only the term involving \( P_n(x) \). This feature is why using orthogonal polynomials like Legendre polynomials is very efficient for series expansions and solving differential equations.