Questions tagged [special-functions]
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58
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Expansion of interaction potential in terms of spherical harmonics in unconventional superconductivity
I am currently reading the book Introduction to Unconventional Superconductivity by V. P. Mineev and K. V. Samokhin. In Equation (1.6), the interaction potential $V$ is expanded in terms of spherical ...
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Complex expression involving pochammer symbols and sums arising from laguerre integral, can it be simplified? [duplicate]
I am working on simplifying a complex expression that arises from a quantum mechanics problem involving matrix elements. The expression is given by an integral involving Laguerre polynomials, which I ...
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1
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Function with two complex variables [closed]
I have a project in an advanced mathematical methods lecture regarding analyticity of functions with two complex variables. My question is, are there some interesting/special functions in $\mathbb C^2$...
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How relationship between the Euler beta function and the strong nuclear force can be mathmatically be proved?
I'm Korean highschool student and was writing a report about Euler beta function and string theory. And I can know find that Euler beta function is similar with the strong nuclear force equation. But ...
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Bosonic Ladder Operators and Coordinate Transformation
What happens the Ladder operators when the problem includes cylindrical symmetry? For example, the energy eigenstates are complicated and Ladder operators change.
Each function requires its own Ladder ...
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Derivation of the Bessel function representation of the Green function of the inhomogeneous Klein-Gordon equation
I will link the following question, as it is partly related to the problem I am trying to deal with.
Green's function for the inhomogenous Klein-Gordon equation
As you can read from this User´s ...
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0
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31
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Examples of complete bases in spherical coordinates [closed]
The set of functions of a three-dimensional harmonic oscillator is a complete basis.
The functions of the 3D harmonic oscillator in the case l=0:
...
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53
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What expression should be used for the radial wave functions of motion in a Coulomb field in order to use them as a basis?
I would like to find the energy eigenvalues by the matrix method and use the radial functions of motion in the Coulomb field as basis functions. But there are radial functions for the continuous ...
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What is the form of general expression (one expression) for the eigenfunctions of discrete and continuous spectra of motion in the Coulomb potential?
Eigenvalues of motion in the Coulomb potential have a discrete spectrum and a continuous spectrum. The eigenwave functions have the form:
For discrete spectrum radial functions(in mathematica code):
<...
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0
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What is the relation between Chebyshev polynomials and coupled oscillators?
I have been told that Chebyshev polynomials are key for finding the normal modes of oscillations of a linear chain of coupled oscillators, since they are the eigenmodes of the system. However, I ...
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Question on the bounds for finding Fourier coefficients
In Griffit's E&M, when solving Laplace's equation for the potential, he uses the "Fourier trick" on Legendre polynomials, where
my question is, why are the bounds from -1 to 1?
because ...
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189
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Green Function expressed in terms of Hankel function (of the second kind)
I am reading Davies' and Birrell's book on QFT in Curved Spacetimes. In Chapter 2 and specifically in the subchapter 2.7, the authors derive an expression for the Feynman propagator
$$G_F(x,x')=\frac{-...
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$q$-dilogartihm function power series
I was reading the $q$-dilogarithm function (Faddev-Kashaev)
https://arxiv.org/abs/hep-th/9310070
How can I derive $$\frac{1}{\Psi(x)}=\sum_{n=0}^{\infty} \theta^{n} x^{n} /(\theta)_{n}$$
by using the ...
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Normalizing the spherical coordinate wavefunction
I try to normalize the following wave function
$$\psi=C_{n l} e^{-\rho / 2} L_{1}^{2 l+1} Y_{l m}$$
Using the normalization condition
$$ 1 = |C_{nl}|^2 \int_{0}^{\infty} e^{-\rho} \rho^2 \{L_{1}^{2l+1}...
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Do these functions (parabolas with opposite parity) ever occur in physics? [closed]
Let's define
$$f_n(x)=(n-1)\operatorname{Li}_n\left(e^x\right)-x\operatorname{Li}_{n-1}\left(e^x\right)$$
(the function $\operatorname{Li}$ is polylogarithm)
$$f_{even_n}(x)=\frac{f(x)+f(-x)}2$$
$$f_{...
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Spherical Bessel Equation has different forms?
I am trying to thoroughly solve the infinite spherical well potential problem that is introduced in Griffith's Introduction to QM, Chapter 4. To solve the Radial part of the equation, one must solve ...
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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 ...
1
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1
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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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1
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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,...
0
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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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1
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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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1
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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$$
is not zero. Am I ...
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Regarding Rayleigh-Sommerfeld Diffraction Integral
While studying the Rayleigh-Sommerfeld diffraction formula I get the standard result for the following integration given at serial no. 11 under section 8.421 of "Table of Integrals" by Gradshteyn and ...
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1
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Different levels of physical model solvability and why reality doesn't care [closed]
In studying physics, one may get the impression that there exists some underlying or even physical difference between models, which solution - motion of a body, wave function - can be found explicitly,...
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Expanding the Green's function in spherical harmonics
I'm currently working through electrodynamics from Purcell supplemented by Jackson and online notes. I've read up the basic cases demonstrating the method of image charges, using the Green's function ...
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References regarding Green's function on a square domain in 2D
Premise: I know this question would be better suited to MathSE, but since I endeavour to solve a free CQFT on a bounded domain, I'm confident I'll find a more exhaustive answer here.
I'm trying to ...
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1
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What physically determines Bessel functions' orders?
I would like a simple explanation of what, in a physical problem, determines the order of the Bessel functions that describe the solutions. What are dimensional(?) parameters of the system that induce ...
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How to calculate hydrogenic radial wave functions?
I'm learning Atkins' Physical Chemistry, and in chapter 9 I found that I cannot get the hydrogenic radial wave function from the formula the book given.
$$
R_{n,l}(r)=N_{n,l} \rho^l L^{2l+1}_{n+1}(\...
3
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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
$$\mathcal{F}\left[\...
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Defining Bessel functions $J_{v}(x)$ for real positive variable $x$?
I've been reading Zangwill's book. The topic of laplace equation in cylindrical coordinates. He proposes defining $J_{v}(x)$ only for $x\geq 0$. He defines it like that but I wonder if there's a ...
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Resources for special functions [closed]
What's a good book for learning about special functions as an undergraduate? I'd prefer information that would be useful to future work in string theory.
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2
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Manipulating Hypergeometric functions
I have a differential equation:
$$(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}$$
For example here ...
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Factorization of 2d Conformal blocks
In 2d CFT, we usually say that the contribution of $z$ and $\bar{z}$ factorize, and then we just consider the $z$ part. In what sense is this true? For example, can the 4-point function be factorized ...
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Imaginary argument in bessel function for a wavefunction
I am solving for the continuum model of haldane model with one of the site being a potential well. The Dirac equation for a topologically non trivial case gives a solution for the states in the band ...
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1
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How are spherical harmonics useful outside class? [closed]
I've learned about spherical harmonics (Legendre polynomials $\longrightarrow$ Associated Legendre polynomials $\longrightarrow$ orthogonality relations $\longrightarrow$ normalization coefficient(s) $...
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To what extent can one recover plane waves from the Airy eigenfunctions of a linear potential as the field is turned off?
Consider a single massive particle in one dimension under the action of a static linear potential, with the hamiltonian
$$
\hat H=\frac{\hat p^2}{2}+\hat{x}F_0.
$$
The eigenstate at energy $E$ is, ...
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Book on gamma functions with applications in physics
I have heard that in my next semester, our quantum mechanics teacher will be giving a great emphasis on difficult integrals with the most of them having to do with gamma functions.
Does anybody know ...
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2
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711
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Legendre Diff. Eq. Appearing in Polar Equation of Hydrogen Atom
The usual form of Legendre's differential equation which I am familiar with, is:
$$
\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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