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Legendre Polynomials

Legendre Polynomials（专名或术语）

One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Legendre polynomials. They are solutions to a very important differential equation, the Legendre equation:

在解决物理问题时遇到的特殊函数之一是被称为勒让德多项式的函数类。它们是解一个非常重要的微分方程——勒让德方程：

The polynomials may be denoted by Pn(x) , called the Legendre polynomial of order n. The polynomials are either even or odd functions of x for even or odd orders n. The first few polynomials are shown below.

这些多项式可表示为Pₙ(x)，称为阶为n的勒让德多项式。这些多项式对于偶数阶或奇数阶n而言，要么是x的偶函数，要么是奇函数。下面列出了前几个多项式。

The general form of a Legendre polynomial of order n is given by the sum:

阶为n的勒让德多项式的通用形式由以下和式给出：

From the Legendre polynomials can be generated another important class of functions for physical problems, the associated Legendre functions.

The equation takes its name from Adrien Marie Legendre (1752-1833), a French mathematician who became a professor in Paris in 1775. He made important contributions to special functions, elliptic integrals, number theory, and the calculus of variations. (Kreyszig).

该方程以法国数学家阿德里安-马里·勒让德（1752-1833）命名。他于1775年在巴黎成为教授，对特殊函数、椭圆积分、数论和变分法等领域做出了重要贡献。（Kreyszig）
从勒让德多项式可以生成另一类重要的函数，用于物理问题，即关联勒让德函数。
 Index

Reference
Kreyszig
Sec 4.3
索引参考Kreyszig第4.3节
 
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Associated Legendre Functions

关联勒让德函数

An important class of special functions called the associated Legendre functions can be derived from the Legendre polynomials. The defining relationship is

一个重要类别的特殊函数，称为关联勒让德函数，可以从勒让德多项式导出。其定义关系为

where Pn(x) is the Legendre polynomial of order n. These functions are of great importance in quantum physics because they appear in the solutions of the Schrodinger equation in spherical polar coordinates. In that context the variable x is replaced by cosq, where q is the colatitude angle. Also in that context, the wavefunctions which are the solutions of the Schrodinger equation need to be normalized, so the list of functions below will include the normalization factor

其中Pn(x)是阶为n的勒让德多项式。这些函数在量子物理中具有重要意义，因为它们出现在球坐标系中施罗德方程的解中。在此背景下，变量x被替换为cosθ，其中θ是极角。在此背景下，施罗德方程的解所对应的波函数需要进行归一化，因此下面列出的函数列表将包括归一化因子。

The normalized functions are of the form

归一化函数的形式为

and appear in the wavefunctions for the hydrogen atom. The first few normalized associated Legendre functions are listed below.

并且出现在氢原子的波函数中。下面列出了前几个归一化的关联勒让德函数。

The associated Legendre functions can be used to construct another important set of functions, the spherical harmonics.

关联勒让德函数可以用来构造另一组重要的函数，即球面谐函数。

 Index

References
Kreyszig
Sec 4.3

Leighton
Ch 5
索引参考 Kreyszig 第4.3节 Leighton 第5章
 
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