正交多项式是一种数学概念，指的是一组满足特定条件的多项式函数。它们在数学和物理学中都有重要的应用，可以用来描述和解决各种问题。

Orthogonal polynomials, also known as orthogonal functions, are a set of polynomial functions that satisfy certain conditions. They have important applications in mathematics and physics, and can be used to describe and solve various problems.

怎么读（音标）

正交多项式 [zhèng jiāo duō xi shì]

用法

正交多项式可以用来表示不同变量之间的关系，也可以作为基函数来展开其他函数。它们具有良好的性质，在数值计算和近似计算中有广泛的应用。

Orthogonal polynomials can be used to represent relationships between different variables, or as basis functions to expand other functions. They possess good properties and have wide applications in numerical and approximation calculations.

例句1-5句且中英对照

1. 正交多项式在物理学中被广泛应用，如量子力学中的谐振子问题。

Orthogonal polynomials are widely used in physics, such as in the harmonic oscillator problem in quantum mechanics.

2. 在数值计算中，正交多项式可以用来构造高效的数值积分方法。

In numerical calculations, orthogonal polynomials can be used to construct efficient numerical integration methods.

3. 正交多项式具有互相正交和归一化的性质，在解决微分方程和最小二乘问题时十分有用。

The orthogonality and normalization properties of orthogonal polynomials are useful in solving differential equations and least squares problems.

4. 勒让德多项式是一种常见的正交多项式，它们在物理学中常用来描述球面上的问题。

Legendre polynomials are a common type of orthogonal polynomials, often used in physics to describe problems on spheres.

5. 贝塞尔函数可以表示为勒让德多项式的线性组合，因此也可以看作是一种正交多项式。

Bessel functions can be expressed as linear combinations of Legendre polynomials, and can therefore also be considered as orthogonal polynomials.

同义词及用法

正交多项式也被称为正交函数，它们具有相似的性质和应用。在不同的领域中，可能会使用不同的名称来指代正交多项式，但它们都指代同一类函数。

Synonyms for orthogonal polynomials include orthogonal functions, which have similar properties and applications. In different fields, different names may be used to refer to orthogonal polynomials, but they all refer to the same type of functions.

编辑总结

正交多项式是一组满足特定条件的多项式函数，在数学和物理学中都有重要的应用。它们具有良好的性质，在数值计算和近似计算中发挥着重要作用。熟悉并掌握正交多项式的概念和应用，可以帮助我们更好地理解和解决各种数学和物理问题。