# Legendre Polynomial Integration

• member 428835
In summary, the task is to simplify the given integral, which involves the Legendre Polynomials and is defined on the interval [-1,1]. The first step is to use integration by parts, and then to express the remaining terms in a simpler manner. The resulting integral can then be further simplified using Legendre's differential equation.
member 428835

## Homework Statement

Simplify $$\int_{-1}^1\left( (1-x^2)P_i''-2xP'_i+2P_i\right)P_j\,dx$$
where ##P_i## is the ##i^{th}## Legendre Polynomial, a function of ##x##.

## The Attempt at a Solution

Integration by parts is likely useful?? Also I know the Legendre Polynomials are orthogonal on ##[-1,1]##. Before simply trying integration by parts term-wise I was trying to see if the equi-dimensional equation ##(1-x^2)P_i''-2xP'_i+2P_i## could first be expressed in a more simple manner? Any thoughts?

Just realized ##(1-x^2)P_i''-2xP'_i = ((1-x^2)P'_i)'##, and obviously the last term ##2P_i## instantly vanishes when ##i\neq j##. Then the above integral becomes $$(1-x^2)P'_iP_k|_{-1}^1-\int_{-1}^1(1-x^2)P'_iP'_k\,dx+\frac{2}{2i+1}\delta_{ik}$$ where I use the kronicker delta. Does this simplify further?

Edit: yes it does further simplify. Look at Legendre's DE. It's fairly straightforward from here on out.

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## What is Legendre Polynomial Integration?

Legendre Polynomial Integration is a mathematical technique used to approximate the definite integral of a function using Legendre polynomials. It is a useful tool in numerical analysis and is commonly used in many branches of science and engineering.

## How does Legendre Polynomial Integration work?

Legendre Polynomial Integration involves approximating the definite integral of a function by representing it as a linear combination of Legendre polynomials. The coefficients of the polynomials are determined using the method of least squares, which minimizes the errors between the actual function and the polynomial approximation.

## What are the advantages of using Legendre Polynomial Integration?

Legendre Polynomial Integration has several advantages, including its ability to handle a wide range of functions, its high accuracy, and its ease of implementation. It also has the added benefit of being computationally efficient, making it a popular choice for numerical integration.

## What are the limitations of Legendre Polynomial Integration?

While Legendre Polynomial Integration is a powerful tool, it does have some limitations. It may not be suitable for functions with highly oscillatory behavior or those with discontinuities. It is also not well-suited for integrals over unbounded domains. In these cases, other numerical integration methods may be more appropriate.

## How is Legendre Polynomial Integration used in scientific research?

Legendre Polynomial Integration is commonly used in scientific research to approximate integrals that cannot be solved analytically. It is particularly useful in fields such as physics, engineering, and statistics, where numerical integration is often necessary. It allows scientists to accurately and efficiently solve complex integrals, leading to a better understanding of the underlying phenomena.

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