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Tagged with legendre-polynomials bessel-functions
11
questions
1
vote
2
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289
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The $n$th derivative of the $n$th spherical Bessel function
I quote Problem 12.4.7 of the 5th edition of Mathematical Methods for Physicists by Arfken, Weber, and Harris:
A plane wave may be expanded in a series of spherical waves by the Rayleigh equation:
$$
...
- 347
4
votes
1
answer
171
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Proving $\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}\big(P_{2l}(x)+\text{sgn}(x)P_{2l+1}(x)\big)$
Can someone help me in proving the following:
$$
\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}(P_{2l}(x)+\text{sgn}(x)\cdot P_{2l+1}(x)),
$$
for any value of $x$, $-1\le x\le 1$? (Here $P_l(x)$ ...
3
votes
1
answer
176
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Convert ODE to a form of Bessel differential equation
I'm working on the solution of the equation
$$\tan^2u\partial^2_u y_2 + (2+\tan^2u)\tan u \partial_u y_2 -a^2\lambda_2y_2 - n^2(1+\cot^2u)y_2 = 0.$$
It is possible to write the above equation in terms ...
- 373
3
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1
answer
247
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How to relate $\int_{-1}^{1}e^{ikru} P_{\ell}(u)du$ to $j_\ell(kr)$ in a simple way?
Consider the following epansion of the function $e^{ikru}$ in terms of Legendre polynomials, $P_\ell(u)$, $$e^{ikru}=\sum_{\ell=0}^{\infty}C_\ell(r)P_\ell(u)$$ where $k$ is a constant real parameter, ...
- 693
18
votes
0
answers
884
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How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
By experimenting in Mathematica, I have found the following expression for the integral:
$$
\int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}...
- 469
0
votes
0
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137
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Integral of Bessel functions multiply Legendre polynomials.
Given $x_{i}$ is a fixed point in [-L,L], $l_{0}$ and $L$ is also positive constant.
$|x_{i}-y|$ represent absolute value of $(x_{i}-y)$. define $z=\frac{|x_{i}-y|}{l_{o}}$.
$\mathbf{H_{\alpha}}$ is ...
- 461
3
votes
0
answers
316
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Closed form solution of a contour integral involving Bessel function integral representation
I have the following integral for which I want a closed-form solution:
$$
I_n(kr,\cos\theta) = \int_{C} \frac{e^{ikrt}P_n(t)}{t-\cos\theta} dt,
$$
where $P_n$ is a Legendre polynomial and $C$ is a ...
- 131
1
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0
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130
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Definite Integral in Legendre polynomial formula
\begin{equation*}
\int_{-a}^{a}e^{-\rho x}P_n \left( \frac{x}{a} \right) dx = (-1)^n\sqrt{\frac{2\pi a}{\rho}}J_{n+1/2}(a\rho)
~~~~~~\text{for $a>0$}
\end{equation*}
I am just wondering this ...
- 21
0
votes
1
answer
329
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Second order ODE bessel function?
I'm currently trying to find the solutions to this second order ODE:
R'' + R'/r + c*R = 0, where R(r) and c is an arbitary constant.
I found this page https://physics.stackexchange.com/questions/...
- 101
3
votes
0
answers
133
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Expansion of some singular kernel with the help of Bessel and Neumann spherical harmonic functions
With the following notations:
$j_n$: spherical Bessel functions,
$y_n$: spherical Neumann function,
$P_n$: Legendre polynomial,
$r$, $\rho$, $\theta$, $\lambda$ arbitrary complex,
$R=\sqrt{r^2+\rho^2-...
- 485
1
vote
1
answer
682
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I need help finding a generating function using some relation between the Bessel function and the Laplace integral for the Legendre Polynomials
The Bessel function of the first kind and order n has the integral representation
$J_n(z)=i^{-n}/\pi \int_0^\pi e^{iz\cos\theta}\cos(n\theta)d\theta$
By using the Laplace integral for the Legendre ...
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