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Legendre expansion is a powerful tool that expresses a function as a series of orthogonal polynomials. In particular, we use this method to expand functions like absolute value functions in terms of Legendre polynomials within a given interval, such as (-1, 1). The Legendre polynomials, denoted as \( P_n(x) \) , form a complete set of orthogonal functions on this interval.

To perform a Legendre expansion, we start by identifying the function f(x) to expand. The general series is given by:

\[ f(x) = \sum_{n=0}^{∞} a_n P_n(x)\]where a_n are coefficients determined by integrating the product of f(x) and the Legendre polynomial P_n(x) over the interval. This expansion is quite useful because Legendre polynomials have orthogonality properties that make integration simpler, helping us to more easily find these coefficients.

In a Legendre polynomial expansion, finding the coefficients \( a_n \) is a crucial step. These coefficients dictate how much each polynomial contributes to the series representation of a given function.

For Legendre polynomials, the coefficients can be calculated using:\[a_n = \frac{2n+1}{2} \int_{-1}^{1} f(x) P_n(x) dx\]This formula derives from the orthogonality of Legendre polynomials over the interval (-1, 1), which means each polynomial integral with another polynomial of a different order is zero. This orthogonality helps isolate and calculate each coefficient separately.

For example, if f(x) = |x| in the equation, the function has a symmetry around zero. As a result, the coefficients for even n are zero due to the antisymmetric nature of the integrals corresponding to even orders. Only odd-order polynomials contribute, simplifying the calculation of the series.

Integral calculation is a core part of obtaining coefficients in orthogonal polynomial expansions.

When dealing specifically with Legendre polynomials, we are tasked with solving integrals of the form:\[a_n = \frac{2n+1}{2} \left( \int_{-1}^{0} -x P_n(x) dx + \int_{0}^{1} x P_n(x) dx \right)\]Here, the function f(x) = |x| splits the integral into two parts because f(x) behaves differently over negative and positive intervals. This split is necessary because |x| represents x when x > 0 and -x when x < 0.

The solution involves evaluating each integral separately and then combining them. This method ensures the properties of the function are adequately represented over the full interval. Solving these integrals requires knowledge of calculus and properties of Legendre polynomials, but the orthogonality helps by nullifying some terms automatically, particularly simplifying calculations for even orders in symmetric functions like |x|.