Questions tagged [special-functions]

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Why for cylindrical problems the order of the Bessel function is an integer value?

I read somewhere that for cylindrical problems the order of the Bessel function is an integer value while for spherical problems the order is of half integer value. I know that the Bessel functions ...
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Why do Hankel functions represent travelling waves?

I was trying to find the fundamental solutions for the Helmholtz equation in $\mathbb{R}^d$ when I found this answer. Here, and in some other places, it is stated that Hankel functions represent ...
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Indices In The Hyperspherical Harmonics

Definition In $d$-dimensional space we have a hyperspherical coordinate system with angles $\theta_1, \theta_2, ..., \theta_{d-2}, \phi$. I am working with the following definition (up to ...
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What does a star on a spherical harmonic mean?

So I was studying multipole expansion and the book I am using introduced spherical harmonics. While I could understand the concept of the functions themselves, the book suddenly started putting a “ * “...
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What does it mean that Zernike's polynomials form a orthogonal basis on the unit disk?

I am doing a project on adaptive optics and I would like to understand a little more about Zernike's polynomials. What does it mean that they form an orthogonal basis on the unit circle? What ...
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Orthonormalization condition for $L^2$ operator

This is ortho-normalization condition for eigenfunctions of square of orbital angular momentum: $$\int_0^{2\pi}d\phi\int_0^\pi d\theta\sin\theta \ Y^*_{l'm'}(\theta,\phi) Y_{lm}(\theta,\phi)=\delta_{l,...
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Deriving recurrence of the Hermite polynomials

I am trying to follow Sakurai in Modern Quantum Mechanics, 3rd Ed., Section 2.5. We define the Hermite polynomials as $$ g(x,t)\equiv e^{2xt-t^2}=\sum_{n=0}^\infty\frac{(2xt-t^2)^n}{n!} =\sum_{n=0}^\...
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Spherical Harmonics Sum Identity

I'm taking a course in Quantum Mechanics and this problem is causing me some struggles. Can anyone help me to prove this identity? $$\sum_{m = -l}^l m^2 |Y_{l}^{m}(\theta, \phi)|^2 = \frac{l(l+1)(2l+1)...
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Is there a way to get the generating function of Hermite polynomials?

I would like to know if there is any physical model in which the generating function of the Hermite polynomials arises, I know the problem of the quantum harmonic oscillator but I have not found the ...
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Why does Jackson's book use different method to calculate the vectorial multiple expansion coefficients in chapter 10.3 and 9.7?

In chapter 9.7: $$Z_0a_E(l,m)f_l(kr)=-\frac{k}{\sqrt{l(l+1)}}\int{Y^*_{lm}\mathbf{r\cdot E}d\Omega}\tag{9.123}$$ In chapter 10.3 $$a_\pm(l,m)j_l(kr)=\int{\mathbf X^*_{lm}\mathbf\cdot\mathbf E(\mathbf ...
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How fast should you rotate a chain hoop so that it doesn't tilt?

Consider spinning a closed chain as in the figure above, I want the speed at which the chain will be kept in a horizontal plane. Inspired by this irodov question Considering taking the point where the ...
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What is the general solution of the Associated Legendre differential equation when $A$ does not equal $\ell(\ell+1)$?

I am sorry if this question isn't clear, I couldn't think of a better way to phrase it. I am a Physics student trying to solve the angular component of the wave function for a particle in a central ...
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Addition theorem for Spherical Bessel function

Is there a similar version of the summation theorem for spherical Bessel functions $j_{k}$ that of standard Bessel function $J_{k}$? Especially look below Eq.(3) in this paper https://arxiv.org/abs/...
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Diffusion on a circle?

I've been trying to solve the diffusion equation on a circle. The problem I am running into is that because of the periodic boundary, the wavevector k (when you Fourier transform) gets quantized ...
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Integral of Hermite functions

In the treatment of the quantum harmonic oscillator appear integrals like \begin{equation*} \int_{-\infty}^{+\infty} \mathrm d \zeta \; e^{-\zeta^2} H_{n}(\zeta+\zeta_1) H_{m}(\zeta+\zeta_2) \end{...
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Is there a generating function for Hermite polynomials of 2n?

I want to know if exists a generating function for the hermite polynomials that is $H_{2n}(x)$
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Why are some associated Legendre functions not orthogonal to each other?

For example, $P_1^1 = sin\theta$ and $P_2^2 = 3 sin^2\theta$ seem to be not orthogonal to each other because the integral $$\int_0^\pi (sin\theta)(3 sin^2\theta) sin\theta d\theta$$ is not zero. Am I ...
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Regarding Rayleigh-Sommerfeld Diffraction Integral

While studying the Rayleigh-Sommerfeld diffraction formula I get the standard result for the following integration given at serial no. 11 under section 8.421 of "Table of Integrals" by Gradshteyn and ...
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Different levels of physical model solvability and why reality doesn't care [closed]

In studying physics, one may get the impression that there exists some underlying or even physical difference between models, which solution - motion of a body, wave function - can be found explicitly,...
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Expanding the Green's function in spherical harmonics

I'm currently working through electrodynamics from Purcell supplemented by Jackson and online notes. I've read up the basic cases demonstrating the method of image charges, using the Green's function ...
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References regarding Green's function on a square domain in 2D

Premise: I know this question would be better suited to MathSE, but since I endeavour to solve a free CQFT on a bounded domain, I'm confident I'll find a more exhaustive answer here. I'm trying to ...
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What physically determines Bessel functions' orders?

I would like a simple explanation of what, in a physical problem, determines the order of the Bessel functions that describe the solutions. What are dimensional(?) parameters of the system that induce ...
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How to calculate hydrogenic radial wave functions?

I'm learning Atkins' Physical Chemistry, and in chapter 9 I found that I cannot get the hydrogenic radial wave function from the formula the book given. $$ R_{n,l}(r)=N_{n,l} \rho^l L^{2l+1}_{n+1}(\...
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On fusion transformation in Liouville CFT

It is known in Liouville CFT from the crossing symmetry that the four points $s$-channel and 4t$-channel conformal blocks are related to each other via an integral transformation $$\mathcal{F}\left[\...
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Defining Bessel functions $J_{v}(x)$ for real positive variable $x$?

I've been reading Zangwill's book. The topic of laplace equation in cylindrical coordinates. He proposes defining $J_{v}(x)$ only for $x\geq 0$. He defines it like that but I wonder if there's a ...
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Resources for special functions [closed]

What's a good book for learning about special functions as an undergraduate? I'd prefer information that would be useful to future work in string theory.
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Manipulating Hypergeometric functions

I have a differential equation: $$(f\Phi')'-\frac{l(l+1)}{r^2}\Phi=0,$$ which I've solved with a program that yields $$\Phi(r)=C_1 (\frac{r}{R})^{2}{_2 }F_1 (1-l,2+l,3,\frac{r}{R}).$$ While this ...
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Why do we need the Condon-Shortley phase in spherical harmonics?

I'm confused with different definitions of spherical harmonics: $$Y_{lm}(\theta,\phi) = (-1)^m \left( \frac{(2l+1)(l-m)!}{4\pi(1+m)!} \right)^{1/2} P_{lm} (\cos\theta) e^{im\phi}$$ For example here ...
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Factorization of 2d Conformal blocks

In 2d CFT, we usually say that the contribution of $z$ and $\bar{z}$ factorize, and then we just consider the $z$ part. In what sense is this true? For example, can the 4-point function be factorized ...
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Imaginary argument in bessel function for a wavefunction

I am solving for the continuum model of haldane model with one of the site being a potential well. The Dirac equation for a topologically non trivial case gives a solution for the states in the band ...
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How are spherical harmonics useful outside class? [closed]

I've learned about spherical harmonics (Legendre polynomials $\longrightarrow$ Associated Legendre polynomials $\longrightarrow$ orthogonality relations $\longrightarrow$ normalization coefficient(s) $...
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13 votes
3 answers
232 views

To what extent can one recover plane waves from the Airy eigenfunctions of a linear potential as the field is turned off?

Consider a single massive particle in one dimension under the action of a static linear potential, with the hamiltonian $$ \hat H=\frac{\hat p^2}{2}+\hat{x}F_0. $$ The eigenstate at energy $E$ is, ...
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1 answer
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Book on gamma functions with applications in physics

I have heard that in my next semester, our quantum mechanics teacher will be giving a great emphasis on difficult integrals with the most of them having to do with gamma functions. Does anybody know ...
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Legendre Diff. Eq. Appearing in Polar Equation of Hydrogen Atom

The usual form of Legendre's differential equation which I am familiar with, is: $$ \left(1-x^2\right)\frac{\mathrm d^2P}{\mathrm dx^2} - 2x\frac{\mathrm dP}{\mathrm dx} + \ell\left(\ell+1\right)P = 0 ...
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2 votes
0 answers
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Why does regularization work in this Bessel function integral?

I encountered some days before an integral representation for a modified Bessel function and should differentiate it. But in this representation : $$K(\omega,a)=\int_0^{\infty} \frac{ds}{s} e^{-i\...
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I want to solve Mathieu Equation $y''(x)+(a−2q \cos(2x))y(x)=0$. How to solve it using Floquet solution?

I want to solve Mathieu Equation $$y''(x)+(a−2q \cos(2x)) \, y(x)=0.$$ How to solve it using Floquet solution? In Floquet solution for integer order of $v$ and $π$ periodicity We have Solution $$y(x)=\...
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3 votes
1 answer
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Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
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8 votes
1 answer
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What's the relation between the Euler $\psi$ function, the digamma function, and the hypergeometric function?

Can somebody help me out with the intermediate details of eqn. (2.5) in this paper? Generalized gravitational entropy. Aitor Lewkowycz and Juan Maldacena. arXiv:1304.4926. Is the Euler $\psi$ ...
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Why can the Euler beta function be interpreted as a scattering amplitude?

The Wikipedia article on the Veneziano Amplitude claims that the Euler beta function can be interpretted as a scattering amplitude. Why is this? In another word, when the Euler beta function is ...
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4 votes
3 answers
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Complex energy eigenstates of the harmonic oscillator

Given the Hamiltonian for the the harmonic oscillator (HO) as $$ \hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,, $$ the Schroedinger equation can be reduced to: $$ \left[ \frac{d^2}{dz^2}-\...
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3 votes
1 answer
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Shift operator (integral calculus involving Hermite polynomials) [closed]

I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and ...
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1 answer
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Calculation of spherical Bessel functions - meaning of $\left(\frac{1}{x}\frac{d}{dx}\right)^{l}$

I'm trying to understand the calculation of spherical Bessel functions in chapter four of Griffiths' Introduction to Quantum Mechanics (2nd ed, p142). He gives $$j_{2}\left(x\right)=\left(-x\right)^{2}...
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