All Questions
Tagged with legendre-polynomials recurrence-relations
26
questions
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Derivation of Legendre Polynomials from only orthogonality
I recently stumbled on the idea of Legendre polynomials, from the perspective of using them to approximate functions over a region, and I discovered all of these other ways to express them, using ...
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1
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Getting the recurrence relation for Legendre polynomials by Leibnitz rule
Exercise:
For each natural number $n$ define
$$\phi_n(x)=\frac{d^n}{d x^n}\left(x^2-1\right)^n$$
Derive the formulas
$$\phi_{n+1}^{\prime}(x)=2(n+1) x \phi_n^{\prime}(x)+2(n+1)^2 \phi_n(x)$$
$$\phi_{n+...
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1
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How to prove Legendre Polynomials' recurrence relation without using explicit formula?
Assume there is an inner product on linear space $V = \{ \text{polynormials}\}$:
$$\langle f, g\rangle = \int_{-1}^1 w(t) f(t)g(t) dt$$ with $w(t) \ge 0$ and not identically zero.
Then we can ...
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1
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1
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How can I prove differential recurrence relation of Legendre polynomials without generating function?
What I am trying to prove is $P'_{n+1}(x)−P'_{n−1}(x)=(2n+1)P_n(x)$
What I can use are here:
$P_0(x)=1$, $P_1(x)=x$
$\int_{-1}^1 P_m(x)P_n(x) \;dx = 0 \;(n\neq m)$ (the orthogonality of Legendre ...
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1
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148
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recurrence relation associated Legendre functions
I need a little help to find the recurrence relation
$$\sqrt{1-x^2}P_l^m(x) = \frac{1}{2l+1} (P_{l-1}^{m+1}-p_{l+1}^{m+1})$$
Using the identity
$$(2l+1)P_l(x) = \frac{d}{dx}(P_{l+1}(x)-P_{l-1}(x))$$
I ...
- 47
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Legendre polynomials satisfying a recurrence relation.
Monic Legendre polynomials (which are orthogonal polynomials) on $[-1,1]$ are defined as follows:
$$p_{0}(x)=1, \ p_{1}(x)=x$$
and $p_{n}(x)$ is a monic polynomial of degree $n$ such that $$\int_{-1}^{...
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117
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Alternative definition of Legendre polynomials
I'm studying Panofsky and Phillips' Classical Electricity and Magnetism. In writing the potential of a linear $2^n$-pole lying along the $x$-axis, they make use of the following definition for the ...
- 4,119
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302
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Associated Legendre Polynomials recurrence relations
I am trying to find the following recurrence relation for these polynomials concerning its derivative:
$$(1-x^2)\frac{dP_l^m}{dx}=-lxP_l^m+(l+m)P_{l-1}^m$$
employing the generating function:
$$T_m(...
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1
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Legendre polynomial recurrence relation proof using the generation function
I want to prove the following recurrence relation for Legendre polynomials:
$$P'_{n+1}(x) − P'_{n−1}(x) = (2n + 1)P_n(x)$$
Using the generating function for the Legendre polynomials which is,
$$(1-...
- 519
1
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1
answer
1k
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Finding the Fourier-Legendre series of a function
I need to find the Fourier-Legendre expansion of the function $f(x)=(1-x^2)^{-1/2}$.
The Fourier-Legendre expansion is
$$f(x) = \sum_{n=0}^{\infty}a_nP_n(x)$$
where the coefficients of the series ...
- 27
1
vote
1
answer
238
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Recurrence relations and power series solution
I am given the following initial value problem: $$(1-x^2)y''+7xy'-26y=0 \qquad , \qquad y(0)=0 \qquad , \qquad y'(0)=4$$
I have solved for the singular points, which are $x= 1, -1$
The question ...
- 89
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96
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Orthogonal Property of Legendre Polynomials
How can I get $$nu_{n} + (n-1)u_{n-1},$$ where
$$u_{n} = \int_{-1}^{1} x^{-1}P_{n-1}(x) P_{n}(x)\, \mathrm {d}x\; ? $$
I did many search, and also I did try by myself. But without a success.
Is ...
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0
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411
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Legendre recurrence relation
I am having a slight issue with generating function of Legendre polynomials and shifting the sum of the generating function.
So here is an example:
I need to derive the recurrence relation $lP_l(x)=(...
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2
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1k
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Differential Equation from a generating function related to legendre polynomial
Consider the generating function $$\frac1{1-2tx+t^2}=\sum_{n=0}^{\infty}y_n(x)t^n$$. I wish to find a second order differential equation of the form $$p(x)y_n''(x)+q(x)y_n'(x)+\lambda_ny_n(x)=0$$ and ...
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How to get the closed-from of the series with Legendre polynomials
I want to find the closed-form of the Legendre series of
$S(i,j)=\sum_{n=j}^\infty {\varepsilon}^n n^i P_n^j(\cos \theta)$
with $i$ a arbitray integer, $j$ a postive integer, $\varepsilon$ a real ...
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