mathematics

Legendre polynomials

Legendre polynomials is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Legendre polynomials rather than just read about it. In short: In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applicati…

Key takeaways

Legendre polynomials belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Legendre polynomials to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Legendre polynomials from memory before moving on to harder problems.

Reference excerpt

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions.

Definition and representation

Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w ( x ) = 1 {\displaystyle w(x)=1} over the interval [ − 1 , 1 ] {\displaystyle [-1,1]} . That is, P n ( x ) {\displaystyle P_{n}(x)} is a polynomial of degree n {\displaystyle n} , such that

∫ − 1 1 P m ( x ) P n ( x ) w ( x ) d x = ∫ − 1 1 P m ( x ) P n ( x ) d x = 0 if n ≠ m . {\displaystyle \int _{-1}^{1}P_{m}(x)P_{n}(x)w(x)\,dx=\int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx=0\quad {\text{if }}n\neq m.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Legendre polynomials

Start with the simplest possible case. Write down what Legendre polynomials claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Legendre polynomials before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Legendre polynomials ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Legendre polynomials

In research: Legendre polynomials appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Legendre polynomials in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Legendre polynomials is common in secondary-school and first-year university syllabi. It links to neighbouring topics Orthogonal polynomials, Polynomials, Special hypergeometric functions, so understanding it makes those chapters shorter.
In everyday life: Look for Legendre polynomials outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Legendre polynomials in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Legendre polynomials means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Legendre polynomials out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Legendre polynomials in simple terms?

Why does Legendre polynomials matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Legendre polynomials?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Legendre polynomials.