# Questions tagged [special-functions]

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### Commutator of raising operator in angular momentum with partial derivative wrt z

While fiddling around with certain commutation relations, i noticed the following relation while using spherical coordinates. What could this relation mean intuitively? Let me know if any information ...
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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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1 vote
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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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### 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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1 vote
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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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1 vote
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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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### 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 ...
186 views

### 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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### 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 vote
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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 vote
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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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### 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 ...
685 views

### 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 ...
1 vote
560 views

### 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/...
391 views

### 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 ...
167 views

### 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 ...
1k views

### 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 ...
1 vote
336 views

### 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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### 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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