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# 数学代写|傅里叶分析代写Fourier Analysis代考|Properties of Legendre Polynomials

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Properties of Legendre Polynomials

The Rodrigues Formula.
We CAN Do eXamPLes of Fourier-Legendre eXPansions given just a few facts about Legendre polynomials. The first property that the Legendre polynomials have is the Rodrigues Formula:
$$\operatorname{Pn}(x)=12 n n ! d n d x n(x 2-1) n, n \in N 0 .(3.44)$$
From the Rodrigues formula, one can show that $P_n(x)$ is an $n$th degree polynomial. Also, for $n$ odd, the polynomial is an odd function and for $n$ even, the polynomial is an even function.
Example 3.7. Determine $P_2(x)$ from the Rodrigues Formula.
This is a straightforward application of the Rodrigues Formula:
$$P 2(x)=1222:(2 d 2 x 2(x 2-1) 2=182 d x 2(x+2 x 22+1)=18 d d x(4 x 3-4 x)=18(12 \times 2-4)=12(3 \times 2-1) \cdot(3.45)$$
Note that we obtained the same result as we found in the last section using orthogonalization.

## 数学代写|傅里叶分析代写Fourier Analysis代考|The Differential Equation for Legendre Polynomials

THE LEGENDRE POLYNOMIALS SATISFY a second-order linear differential equation. This differential equation occurs naturally in the solution of initialboundary value problems in three dimensions which possess some spherical symmetry. There are two approaches we could take in showing that the Legendre polynomials satisfy a particular differential equation. Either we can write down the equations and attempt to solve it, or we could use the above properties to obtain the equation. For now, we will seek the differential equation satisfied by $P_n(x)$ using the above recursion relations.
We begin by differentiating Equation (3.66) and using Equation (3.62) to simplify:
$$d d x((x 2-1)) P^{\prime} n((x))=n \operatorname{Pn}(x)+n x P^{\prime} n(x)-n P^{\prime} n-1(x)=n \operatorname{Pn}(x)+n 2 P n(x)=n(n+1) \operatorname{Pn}(x) \cdot(3.70)$$
Therefore, Legendre polynomials, or Legendre functions of the first kind, are solutions of the differential equation
$$(1-x 2) y^{\prime \prime}-2 x y^{\prime}+n(n+1) y=0$$
A generalization of the Legendre equation is given by $(1-x 2) y^{\prime \prime}-2 x y^{\prime}+[n(n+1)-m 21-x 2] y=0$. Solutions to this equation, Pnm(x) and Qnm(x), are called the associated Legendre functions of the first and second kind.

As this is a linear second-order differential equation, we expect two linearly independent solutions. The second solution, called “the Legendre function of the second kind,” is given by $Q_n(x)$ and is not well behaved at $x= \pm 1$.
For example,
$$\mathrm{Qo}(x)=12 \ln 1+\mathrm{x} 1-\mathrm{x}$$
We will not need these for physically interesting examples in this book.

## 数学代写|傅里叶分析代写Fourier Analysis代考|Properties of Legendre Polynomials

$$\operatorname{Pn}(x)=12 n n ! d n d x n(x 2-1) n, n \in N 0 .(3.44)$$

$$P 2(x)=1222:(2 d 2 x 2(x 2-1) 2=182 d x 2(x+2 x 22+1)=18 d d x(4 x 3-4 x)=18(12 \times 2-4)=12(3 \times 2-1) \cdot(3$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|The Differential Equation for Legendre Polynomials

$$d d x((x 2-1)) P^{\prime} n((x))=n \operatorname{Pn}(x)+n x P^{\prime} n(x)-n P^{\prime} n-1(x)=n \operatorname{Pn}(x)+n 2 P n(x)=n(n+1) \operatorname{Pn}(x) \cdot(3.70)$$

$$(1-x 2) y^{\prime \prime}-2 x y^{\prime}+n(n+1) y=0$$

$$\mathrm{Qo}(x)=12 \ln 1+\mathrm{x} 1-\mathrm{x}$$

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