All Questions
Tagged with legendre-polynomials fourier-series
21
questions
3
votes
1
answer
74
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Legendre expansion of a root polynomial
What is the Legendre expansion of $\sqrt{1-x^2}$ ? I need a closed form. Thx.
I have tried to reform it
$$
\sqrt{1-x^2}=\left. \sqrt{1-2tx+x^2} \right|_{t=x}
$$
while the generating function of ...
2
votes
1
answer
567
views
Problem with Legendre-Fourier series for sinx when the number of terms approaches infinity
After I learned about Fourier series expansion, I understand orthogonality of trigonometric functions was the key when I calculate the coefficients of Fourier series. As I knew that Legendre ...
- 23
0
votes
0
answers
400
views
Show that the inner product of the Legendre polynomials and $p\in L^2 ([-1,1])$ is equal to zero if the polynomial has degrees less than $n$
Let $(P_n )^{\infty}_{n=0}$ be the Legendre polynomials in $L^2([-1,1])$, normalised as $||P_n||^2=\frac{2}{2n+1}$.
I need to show that, for any polynomial $p\in L^2([-1,1])$ with degree less than $n$,...
- 174
1
vote
0
answers
127
views
Expand square of an associated Legendre polynomial in terms of simple associated Legendre Polynomials
I have an associated Legendre Polynomial $\left(P_l^m(\cos(\theta))\right)^2$ (where $l$ and $m$ are nonnegative integers). I need to find a way to express it in terms of simple associated Legendre ...
- 167
1
vote
0
answers
173
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Expanding function on [0,1] using Legendre polynomials
A quick question. Given a boundary condition that a function $f(x)=\sum_{n=0}^\infty a_n P_{2n+1}(x)$ , which is defined for $x$ in $[0,1]$, not for $-1$ to $1$.
I know the standard Fourier- Legendre ...
- 195
0
votes
1
answer
213
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Fourier-Legendre series - Need answers for all 5 question marks
Let $f(x)= |x|$ on $-1\leq x\leq 1$.
Then there is a Fourier-Legendre expansion
$f(x)$ = $\sum_{m=0}^{\infty} c_mP_m(x)$ where
$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left[(x^2-1)^n\right]$
$c_0 ...
- 21
5
votes
1
answer
207
views
Showing a summation identity for $1$, possibly tied to Legendre polynomials
The Problem: Consider the sign function on $(-1,0)\cup(0,1)$ defined by
$$ \sigma(x) := \left. \text{sgn}(x) \right|_{(-1,0)\cup(0,1)} = \begin{cases} 1 & x \in (0,1) \\ -1 & x \in (-1,0) \end{...
- 45.9k
3
votes
0
answers
563
views
Why the expansion of functions of 2 variables in legendre polynomials does not take into account all products Pi (x) Pj (y)?
My question arises from this reading link (In this article, in equation 6 on page 3, also expands a function of 2 variables in this way Parametric estimate of intensity inhomogeneities applied to MRI),...
0
votes
1
answer
545
views
Fourier-Legendre series
I am given a function of $\theta$ as $$f(\theta)=\ln\left(\dfrac{1+\sin\frac\theta2}{\sin\frac\theta2}\right)$$
and I am asked to express this function as a Fourier-Legendre series of the form $\sum_{...
- 2,885
0
votes
1
answer
159
views
Find the value $P_n(1)$ of the Legendre polynomials from their generating function
Here is the question:
Using $\sum_{n=0}^{\infty}P_n(x)r^n=(1-2rx+r^2)^{-\frac{1}{2}}$ find the value of $P_n(1)$
I am unsure how to handle this question. I did the following and would like ...
- 395
2
votes
2
answers
339
views
Intuition for polynomial bases
In my linear algebra course I stumbled upon the following observations.
We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$.
$f(x)$ may be composed of elementary functions or not, but in ...
- 21
0
votes
1
answer
53
views
Find a recurrence relationship for the following :
Find a recurrence relationhip for $a_{n}$:
$a_{n}=\dfrac {2n+1}{2}\int^{1}_{-1}f\left( x\right) P_{n}\left( x\right) dx$
Where $f\left( x\right)= e^{-x}$
I have done it many times and keep ...
- 379
0
votes
1
answer
360
views
Find a recurrence relation and the Fourier-Legendre Series
Rodrique's Formula for the $n$th Legendre Polynomial is
$$P_n\left(x\right)=\dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}\left(\left(x^2-1\right)^n\right)$$
The Fourier-Legendre series of a function f is ...
- 597
1
vote
1
answer
226
views
Integrate the Fourier Legendre by parts :$\int_{-1}^{1}\left( x^{2}-1\right) ^{m}\cos \pi x\:dx$
Having difficulty integrating the Fourier Legendre series by parts :
$$\alpha_{m}=\int_{-1}^{1}\left( x^{2}-1\right) ^{m}\cos \pi x\:dx$$
I understand we can use the general formula :
$$uv-\int ...
- 379
2
votes
1
answer
683
views
Use integration by parts to verify the following :
Using integration by parts show that:
$\int^{1}_{-1}P_{n}\left( x\right) P_{m}\left( x\right) dx$ = $\dfrac {2}{2n+1}, m=n$
and
$0$ if $m\neq n$
Where the functions are both Legendre polynomials.
...
- 379