Questions tagged [legendre-polynomials]
For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.
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The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-\left(\frac{-ab}{p}\right)$
What I need to show is that
For $\gcd(ab,p)=1$ and p is a prime, the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $$p-\left(\frac{-ab}{p}\right)\,.$$
I got a hint that ...
- 2,451
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Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$
Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in L^{...
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Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$
The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula: $$\color{blue}{\displaystyle\int_{x=-1}^{1}[{P_{L}}^m(x)]^2\,\mathrm{d}x=\left(\...
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Proving that Legendre Polynomial is orthogonal
This is from here
$$\int^1_{-1}f_n(x)P_n(x)dx = 2(-1)^n\frac{a_n}{2^n}\int^1_0(x^2-1)^ndx=2(-1)^n\frac{a_n}{2^n}.I_n$$........(6)
I don't understand as in shouldnt it be like this, $$\int^1_{-1}f_n(...
- 499
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1
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Legendre Polynomials integral
I've been asked to calculate:
$$
\int_{0}^{1} P_{\ell}(x)dx,
$$
where $P_{\ell}(x)$ is a Legendre polynomial by using:
i)The generating function:
$$
\sum_{\ell=0}^{\infty}P_{\ell}(x)t^{\ell}=\frac{1}{\...
32
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5
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Roots of Legendre Polynomial
I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
Are the roots always simple (i.e., multiplicity $1$)?
...
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25
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3
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An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$
When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\...
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Fourier Legendre expansion for $\frac{\text{Li}_2(x)}{x}$
Background: I'm computing harmonic series using FL expansion. For instance, the following
$\frac{\log (1-x)}{x}=\sum _{n=0}^{\infty } 2 (-1)^{n-1} (2 n+1) P_n(2 x-1) \left(\sum _{k=n+1}^{\infty } \...
- 8,654
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Interleaving of Gaussian quadrature nodes and weights
A Gaussian quadrature is used to approximate the following integral:
$$
\int_{-1}^{1} f(x) dx \approx \sum_{i=1}^n w_i f(x_i).
$$
Numerically I've found an interesting property of $x_i$ and $w_i$: if ...
- 7,563
3
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2
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Integral of Legendre and Chebyshev polynomials.
I am trying to expand Legendre polynomials into Chebyshev polynomials, shown as:
$$P_{n}(x)=\sum_{k=0}^{n}a_{k}T_{k}(x), $$
where $P_{n}$ is Legendre polynomials and $T_{k}$ is Chebyshev polynomials, ...
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Closed form for $S= \sum\limits_{k=0}^n x^k \binom{n}{k}^2$ [duplicate]
I am looking for a closed form for $\displaystyle S= \sum_{k=0}^n x^k \binom{n}{k}^2$. Does there exist such closed form?
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Associated Legendre Polynomials Orthogonality Proof: $\int_{-1}^1 P_k^m(x) \cdot P_l^m(x) \; \mathrm{d} x = \frac{2(l+m)!}{(2l+1)(l-m)!} \delta_{k,l}$
I have to solve the following equation using associated legendre polynomials,
$$\int_{-1}^1 P_k^m(x) \cdot P_l^m(x) \; \mathrm{d} x = \frac{2(l+m)!}{(2l+1)(l-m)!} \delta_{k,l}$$
Where they are ...
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Legendre Polynomial Orthogonality Integral
I want to show that $$\int_{-1}^{1}L_n(x)L_m(x)dx$$ is zero for $m<n$ and $\frac{2}{2n+1}$ for $m=n$. For $m<n$, I want to apply $(x^2-1)^n=(x-1)^n(x+1)^n$ and integration by parts. For $m=n$ it ...
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Evaluate $\int_0^1 x^{a-1}(1-x)^{b-1}\operatorname{Li}_3(x) \, dx$
Define
$\small f(a,b)=\frac1{B(a,b)}\int_0^1 x^{a-1}(1-x)^{b-1} \text{Li}_3(x) \, dx$$ $$=\frac a{a+b}{}_5F_4(1,1,1,1,a+1;2,2,2,1+a+b;1)$
Where $a>-1$ and $b>0$.
$1$. By using contour ...
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Proving Bonnets' Recursion with Rodrigues' Formula
I would like to show that $(n+1)P_{n+1}(x)+nP_{n-1}(x)=(2n+1)xP_{n}(x)$ using Rodrigues' formula, not the generating function.
I got to this point, but have not been able to progress further.
$$(n+1)...
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