Understanding Orthogonal Polynomials in Mathematical Physics

[sniffer]

i am still learning mathematical physics.

i am learning orthogonal polynomials, but still confused.

what is the meaning of "orthogonal" here?

 

[Timbuqtu]

Orthogonality of two elements of a vectorspace (in this case the space of all polynomials) is only defined with respect to an inner product on the space. A possible inner product on a polynomial space could for instance be:

$$\left<f,g\right>= \int_0^1 f(x) g(x) dx$$

Now f and g are orthogonal $$\iff \left<f,g\right>=0$$.

A set of polynomials is called orthogonal if each polynomial is orthogonal to each other polynomial in the set.

 

[M98Ranger]

Orthogonal polynomials are a fundamental concept in mathematical physics that are used to solve a wide range of problems in the field. They are a special type of polynomials that have the property of being orthogonal, meaning they are perpendicular to each other when plotted on a graph. This property allows them to be used in various mathematical equations and calculations to simplify and solve complex problems.

In the context of mathematical physics, orthogonal polynomials are particularly useful because they can be used to represent physical phenomena or systems in a simpler and more efficient manner. For example, they can be used to describe the motion of a particle in a quantum mechanical system or the behavior of a vibrating string.

Furthermore, the orthogonality property of these polynomials also allows for their use in approximation methods, where they can be used to approximate complex functions by a simpler polynomial function. This is particularly useful in numerical methods and simulations in physics.

I understand that learning about orthogonal polynomials can be confusing at first, but with practice and further study, their applications and importance in mathematical physics will become clearer. Keep studying and don't be afraid to ask questions and seek help from your peers or instructors. Good luck in your learning journey!