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Chapter 12: Problem 11

Express each of the following polynomials as linear combinations of Legendre polynomials. Hint: Start with the highest power of \(x\) and work down in finding the correct combination. \(x-x^{3}\)

Short Answer

Expert verified
\(x - x^3 = \frac{5}{2} P_1(x) - \frac{5}{2} P_3(x)\)

Step by step solution

01

Identify the Legendre Polynomials

Legendre polynomials are a sequence of orthogonal polynomials. The first few Legendre polynomials are given by:\[ P_0(x) = 1 \]\[ P_1(x) = x \]\[ P_2(x) = \frac{1}{2}(3x^2 - 1) \]\[ P_3(x) = \frac{1}{2}(5x^3 - 3x) \]
02

Express Each Term Using Legendre Polynomials

The polynomial given is \(x - x^3\). Start with the highest power term and work down:The term \(-x^3\) can be expressed using \(P_3(x)\):\[-x^3 = -\frac{1}{2}(5x^3 - 3x) + \frac{3}{2}x\] Simplify to obtain:\[-x^3 = -\frac{5}{2} P_3(x) + \frac{3}{2} P_1(x)\]
03

Combine All Terms to Form Linear Combination

Add the linear combinations found in the previous step:\[x - x^3 = P_1(x) - (\frac{5}{2} P_3(x) - \frac{3}{2} P_1(x)) \]Combine like terms:\[x - x^3 = \frac{5}{2} P_1(x) - \frac{5}{2} P_3(x)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orthogonal Polynomials
Orthogonal polynomials are a special class of polynomials that are perpendicular to each other according to a specific inner product. This means their product integrates to zero over a certain interval. One of the most famous sets of orthogonal polynomials is the Legendre polynomials, which are particularly important in physics and engineering.

Legendre polynomials, denoted as \(P_n(x)\), follow a specific recurrence relation and are orthogonal on the interval \([-1, 1]\). Each polynomial in the series has a unique power, making them very useful in approximating functions and solving differential equations. The zero integral property over the interval for distinct Legendre polynomials supports their orthogonality.

For example, the first few Legendre polynomials are:
  • \(P_0(x) = 1\)
  • \(P_1(x) = x\)
  • \(P_2(x) = \frac{1}{2}(3x^2 - 1)\)
  • \(P_3(x) = \frac{1}{2}(5x^3 - 3x)\)
Understanding these polynomials and their properties is crucial when dealing with complex problems in various scientific fields.
Linear Combinations
To express a polynomial as a linear combination of other polynomials, we write it as the sum of those polynomials, each multiplied by a coefficient. In the context of Legendre polynomials, any given polynomial can be represented as a sum of terms involving Legendre polynomials.

For example, if we have a polynomial \(x - x^3\), we can express it as a linear combination of Legendre polynomials. This involves matching the degrees of the given polynomial with suitable Legendre polynomials. Starting with the highest power first helps simplify the process. Given the exercise:

The term \(-x^3\) can be expressed using \(P_3(x)\):
\(-x^3 = -\frac{1}{2}(5x^3 - 3x) + \frac{3}{2}x\)
Simplifying: \(-x^3 = -\frac{5}{2} P_3(x) + \frac{3}{2} P_1(x)\)

Thus, combining the terms, we get:

\((x - x^3) = P_1(x) - (\frac{5}{2} P_3(x) - \frac{3}{2} P_1(x))\)

Combine like terms:
\((x - x^3) = \frac{5}{2} P_1(x) - \frac{5}{2} P_3(x)\)

This method shows how any polynomial can be decomposed into simpler, orthogonal components, which is useful for analysis and computation.
Polynomial Transformation
Polynomial transformation involves changing a polynomial from one form to another. When we express polynomials in terms of Legendre polynomials, we transform the standard polynomial basis into an orthogonal basis. This has the advantage of simplifying many calculations, especially those involving integrals.

Legendre polynomial transformations are significant because they help in problems where orthogonality reduces complexity. By transforming polynomials, particularly in fields like quantum mechanics and numerical analysis, simpler and more manageable forms of the functions are obtained.

Given the example from the previous solution, the polynomial \(x - x^3\) is transformed using Legendre polynomials by aligning each part of the original polynomial to the corresponding Legendre polynomial. This transformation allows us to work within an orthogonal basis (Legendre polynomials), simplifying the work needed in solving differential equations and performing integrations.

Using the transformations:
  • Identify the highest-power term and break it down into Legendre components.
  • Move step by step to lower power terms, converting each into the corresponding Legendre format.
  • Combine the results to get the transformed polynomial as a linear combination of Legendre polynomials.
Understanding polynomial transformation is key in advancing in areas that require high precision in functional approximations and solutions.

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Most popular questions from this chapter

A straight wire clamped vertically at its lower end stands vertically if it is short, but bends under its own weight if it is long. It can be shown that the greatest length for vertical equilibrium is \(l,\) where \(k l^{3 / 2}\) is the first zero of \(J_{-1 / 3}\) and $$k=\frac{4}{3 r^{2}} \sqrt{\frac{\rho g}{\pi Y}},$$ \(r=\) radius of the wire, \(\rho=\) linear density, \(g=\) acceleration of gravity, \(Y=\) Young's modulus. Find \(l\) for a steel wire of radius \(1 \mathrm{mm}\); for a lead wire of the same radius.

(a) The generating function for Bessel functions of integral order \(p=n\) is $$ \Phi(x, h)=e^{(1 / 2) x\left(h-h^{-1}\right)}=\sum_{n=-\infty}^{\infty} h^{n} J_{n}(x) $$ By expanding the exponential in powers of \(x\left(h-h^{-1}\right)\) show that the \(n=0\) term is \(J_{0}(x)\) as claimed. (b) Show that $$ x^{2} \frac{\partial^{2} \Phi}{\partial x^{2}}+x \frac{\partial \Phi}{\partial x}+x^{2} \Phi-\left(h \frac{\partial}{\partial h}\right)^{2} \Phi=0 $$ Use this result and \(\Phi(x, h)=\sum_{n=-\infty}^{\infty} h^{n} J_{n}(x)\) to show that the functions \(J_{n}(x)\) satisfy Bessel's equation. By considering the terms in \(h^{n}\) in the expansion of \(e^{(1 / 2) x\left(h-h^{-1}\right)}\) in part (a), show that the coefficient of \(h^{n}\) is a series starting with the term \((1 / n !)(x / 2)^{n}\). (You have then proved that the functions called \(J_{n}(x)\) in the expansion of \(\Phi(x, h)\) are indeed the Bessel functions of integral order previously defined by (12.9) and (13.1) with \(p=n\).)

Express each of the following polynomials as linear combinations of Legendre polynomials. Hint: Start with the highest power of \(x\) and work down in finding the correct combination. \(x^{4}\)

Solve the differential equations by the Frobenius method; observe that you get only one solution. (Note, also, that the two values of \(s\) are equal or differ by an integer, and in the latter case the larger \(s\) gives the one solution.) Show that the conditions of Fuchs's theorem are satisfied. Knowing that the second solution is \(\ln x\) times the solution you have, plus another Frobenius series, find the second solution. $$x^{2} y^{\prime \prime}+\left(x^{2}-3 x\right) y^{\prime}+(4-2 x) y=0$$

Show that the functions \(e^{i n \pi x / l}, n=0,\pm 1,\pm 2, \cdots,\) are a set of orthogonal functions on \((-l, l)\).
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