勒让德方程的解

The Solution of Legendre Equation

勒让德方程两个线性无关的解,分别称为第一类和第二类勒让德函数。在微分方程取本征值情况下,第一类勒让德函数中断为多项式,因此自变量可取任意值(无穷大除外);第二类勒让德函数仍然为无穷级数,当自变量等于±1时发散,绝对值大于1时收敛。由于勒让德方程属于超比方程类型,给出此类型方程不同特殊函数的任意阶导数表达式。在此基础上直接给出第一类勒让德函数的超比表达式,及其与其他特殊函数的理论关系;鉴于求解第二类勒让德函数的复杂性,利用级数展开方法,直接给出第二类勒让德函数的超比表达式。

The two linearly independent solutions of Legendre equation are called the first and second kind of Legendre functions,respectively.When the differential equation takes the eigenvalue,the first kind of Legendre function is interpreted as a polynomial,so the independent variable can take any value(except infinity).The second kind of Legendre function is still infinite series,diverging when the independent variable is equal to±1 and converging when the absolute value is greater than 1.Since Legendre equation belongs to the type of hypergeometric equation,we give the expression of arbitrary order derivatives of different special functions of this type equation.Therefore,the hypergeometric expression of the first kind of Legendre function and its theoretical relationship with other special functions are given directly.In view of the complexity of solving the second kind of Legendre function,the hypergeometric expression of the second kind of Legendre function is directly given by using the series expansion method.