# Questions tagged [legendre-polynomials]

For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.

598 questions
Filter by
Sorted by
Tagged with
1 vote
37 views

35 views

### Legendre polynomial as basis for finite element method

When people say use Legendre polynomial as basis polynomial in fem, are they using the polynomials themselves or the integral of those polynomials? I'm asking this because I have this poisson equation:...
• 191
40 views

### Confusion about spherical harmonics, Legendre polynomials

I'm quite new to the ideas behind spherical harmonics and Legendre polynomials. I have a couple of questions about them. Spherical harmonics, as I understand them, are functions that can be used to ...
23 views

### How to integrate products of Legendre functions over the interval [0,1]

The associated Legendre polynomials are known to be orthogonal in the sense that $$\int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l}$$ This is intricately linked to ...
• 393
1 vote
18 views

• 29
29 views

1 vote
74 views

### Prove that $\sum_{y=0}^{p-1}\left(\frac{y}{p}\right)\left(\frac{y+d}{p}\right)=-1$

The question is from The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $p-\left(\frac{-ab}{p}\right)$ And I wonder how to prove the following equation although someone gives the trick: \begin{...
• 369
58 views

70 views

### Reference(book or article) for an explicit formula of Legendre polynomials

The following explicit formula is stated for Legendre polynomials on Wikipedia. \begin{equation} P_n(x)=\sum_{k=0}^n {n\choose k}{n+k \choose k} \left(\dfrac{x-1}{2}\right)^2 \end{equation} Do you ...
• 45
19 views

### How do we determine if an operator over real functions is normal?

We have the operator $T(f) = (pf')'$, where $p(x) = x^2 - 1$. The inner product is $\displaystyle (f,g) = \int_{-1}^1 f(x)g(x) dx$. How do we infer whether eigenfunctions corresponding to different ...
1 vote
90 views

### Uniqueness of the nodes for Gauss-Legendre quadrature

Gauss-Legendre quadrature approximates $\int_{~1}^{1}f(x)dx$ by $\sum_{i=1}^nw_if(x_i)$. Wikipedia says that This choice of quadrature weights $w_i$ and quadrature nodes $x_i$ is the unique choice ... 1 vote
40 views

### Relationship between Gauss–Legendre quadrature and continued fractions

Wikipedia says the following: Carl Friedrich Gauss was the first to derive the Gauss–Legendre quadrature rule, doing so by a calculation with continued fractions in 1814. https://en.wikipedia.org/... 56 views

• 11