Question

Legendre polynomial \(P_{\mathrm{b}}(x)=\lambda\left(63 x^{5}-70 x^{3}+15 x\right)\) where \(\lambda\) is equal to (a) \(1 / 2\) (b) \(1 / 5\) (c) \(1 / 8\) (d) \(1 / 10\).

Short answer

The value of \( \lambda \) is \( \frac{1}{8} \).

Step-by-step solution

Identify the General Formula

A Legendre polynomial is typically written in the form of \( P_n(x) \), where \( n \) is the degree. We need to compare the given polynomial \( 63x^5 - 70x^3 + 15x \) with the known forms of Legendre polynomials to identify the corresponding value of \( \lambda \).

Recognize Properties of Legendre Polynomials

Legendre polynomials are a set of orthogonal polynomials that eliminate certain terms to fulfill orthogonality. The polynomial object \( 63x^5 - 70x^3 + 15x \) should match the orthogonal Legendre polynomial of the same degree.

Recall Specific Form of P_5(x)

The 5th Legendre polynomial \( P_5(x) \) has the specific form \( \frac{1}{8}(63x^5 - 70x^3 + 15x) \). Comparing this with the given polynomial, we observe both the polynomials are proportional.

Confirm the Constant \( \lambda \)

Given that the provided polynomial and the Legendre polynomial match by a multiplicative constant, \( \lambda \) is exactly the reciprocal of the factor in \( P_5(x) \). Thus, \( \lambda = \frac{1}{8} \).

Key concepts

Orthogonal Polynomials

Orthogonal polynomials are a fascinating concept in mathematics. They form a sequence of polynomials wherein any two different polynomials in the sequence are orthogonal to each other with respect to a given weight function over a certain interval. This basically means that their inner product is zero. Orthogonality is similar to how the x-axis and y-axis are perpendicular in the Cartesian plane.

Some common examples of orthogonal polynomials include Legendre polynomials, Chebyshev polynomials, and Hermite polynomials. Each type of orthogonal polynomial is associated with its own specific properties and orthogonality conditions. For Legendre polynomials specifically, they are orthogonal over the interval [-1, 1] with a weight function of 1, making them particularly simple and popular in various applications, including physics and engineering.

Understanding orthogonal polynomials is essential because they simplify complex problems, like differential equations, by breaking them down into more manageable components. It’s also important to note that orthogonal polynomials form a basis for a function space, allowing any function to be expanded as an infinite series of these polynomials.

Polynomial Degree

The degree of a polynomial is determined by the highest power of the variable present in the polynomial expression. For example, in the polynomial expression given in the exercise, which is a Legendre polynomial, the highest power of the variable \( x \) is 5.

Thus, it is categorized as a 5th-degree polynomial. The degree of a polynomial significantly influences its characteristics. This includes the number of roots it may have, as well as the shape and behavior of its graph.

Identifying the degree is crucial when working with Legendre polynomials as it dictates the form and properties of the polynomial, such as symmetry and specific orthogonality. Each degree corresponds to a distinct Legendre polynomial, like \( P_0(x), P_1(x), ... and P_5(x) \) for the 5th degree, with specific forms that satisfy their orthogonality on the interval \([-1, 1]\).

When comparing polynomials or solving related problems, knowing the degree helps determine solutions and select appropriate polynomials for approximation or fitting tasks.

Mathematical Constants

Mathematical constants are values that are universally fixed and useful across various areas in mathematics. In the context of the exercise, the constant \( \lambda \) plays an essential role in scaling the Legendre polynomial expression.

A crucial mathematical constant in the exercise is the reciprocal of the scaling factor used with the 5th Legendre polynomial. For example, in our solution, a particular constant \( \lambda = \frac{1}{8} \) is required to perfectly align the given polynomial form with its corresponding Legendre polynomial form.

Constants such as \( \pi \), \( e \), and \( \lambda \) have unique properties and applications that allow us to make precise calculations and derive specific results. In mathematical equations, constants provide structure and uniformity.

Understanding the role of these constants is vital as they frequently appear in computations involving physics, engineering, and intuitively in areas like statistical analysis or algorithm design.

Legendre Polynomial Properties

Legendre polynomials possess a range of unique properties that make them indispensable in theoretical and applied mathematics. An interesting feature is their orthogonality, which allows polynomials to be independently balanced over the interval \([-1, 1]\). This symmetry simplifies complex calculations involving integral functions.

Legendre polynomials can be seen as solutions to Legendre’s differential equation, which is pivotal in physics, especially in solving problems involving spherical harmonics. This makes them extremely useful for tackling physical phenomena such as gravitational fields or electromagnetic waves.

Each Legendre polynomial \( P_n(x) \) is defined for a specific degree \( n \), and the sequence starts from zero. They exhibit a recursive relationship, enabling the generation of higher degree polynomials based on lower degree ones. This recursive aspect facilitates the efficient solution of many mathematical problems.

Utilizing the properties of Legendre polynomials in different mathematical and scientific contexts provides powerful insights and tools for approximation, integration, and solving differential equations.