Skip to main content

Questions tagged [special-functions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
1 answer
59 views

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 ...
ilawid's user avatar
  • 51
2 votes
0 answers
41 views

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 ...
caz's user avatar
  • 21
1 vote
1 answer
70 views

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 $\mathbb C^2$...
syphracos's user avatar
  • 127
-1 votes
1 answer
76 views

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 ...
Myj's user avatar
  • 1
4 votes
0 answers
92 views

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 ...
Lorenzo's user avatar
  • 41
1 vote
0 answers
35 views

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: ...
Mam Mam's user avatar
  • 233
1 vote
1 answer
69 views

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): <...
Mam Mam's user avatar
  • 233
2 votes
0 answers
86 views

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 ...
0 votes
0 answers
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 ...
realanswers's user avatar
3 votes
1 answer
322 views

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{-...
schris38's user avatar
  • 3,992
0 votes
0 answers
46 views

$q$-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 ...
Osman ergeç's user avatar
1 vote
0 answers
65 views

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_{...
Anixx's user avatar
  • 11.2k
1 vote
1 answer
125 views

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 ...
Nick Heumann's user avatar
0 votes
1 answer
92 views

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 ...
Paul's user avatar
  • 13
2 votes
0 answers
191 views

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 ...
Manuel Cañizares's user avatar
1 vote
1 answer
74 views

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 ...
Kerr_Max's user avatar
  • 111
0 votes
1 answer
112 views

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 “ * “...
NiceGuy's user avatar
  • 11
1 vote
1 answer
333 views

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 ...
schrodingal's user avatar
-2 votes
1 answer
114 views

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,...
pouya jahanyar's user avatar
0 votes
1 answer
156 views

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}^\...
hodop smith's user avatar
1 vote
1 answer
389 views

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)...
Adán González's user avatar
0 votes
1 answer
462 views

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 ...
Lluis Gerardo's user avatar
0 votes
1 answer
57 views

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 ...
jw_'s user avatar
  • 473
0 votes
2 answers
70 views

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 ...
Cathartic Encephalopathy's user avatar
0 votes
1 answer
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 ...
Kristian Stokkereit's user avatar
1 vote
1 answer
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/...
Boa_Constrictor's user avatar
0 votes
0 answers
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 ...
Mikail Khona's user avatar
0 votes
2 answers
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{...
Graz's user avatar
  • 385
0 votes
2 answers
212 views

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 $H_{2n}(x)$
Henrique Yukio's user avatar
0 votes
1 answer
73 views

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 ...
toby's user avatar
  • 15
2 votes
0 answers
221 views

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 ...
R. Bhattacharya's user avatar
-1 votes
1 answer
117 views

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,...
neverneve's user avatar
  • 773
2 votes
1 answer
912 views

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 ...
abhibrata ganguly's user avatar
3 votes
1 answer
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
1 answer
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 ...
Paul Cwave's user avatar
0 votes
1 answer
2k views

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}(\...
Kane Green's user avatar
3 votes
1 answer
255 views

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[\...
Gropillon's user avatar
0 votes
1 answer
43 views

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 ...
Omar's user avatar
  • 133
1 vote
1 answer
177 views

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.
2 votes
2 answers
204 views

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 ...
Karl's user avatar
  • 207
12 votes
1 answer
4k views

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 ...
ilciavo's user avatar
  • 243
1 vote
1 answer
505 views

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 ...
Nahc's user avatar
  • 2,071
1 vote
0 answers
2k views

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 ...
vbj's user avatar
  • 425
2 votes
1 answer
480 views

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) $...
Arturo don Juan's user avatar
16 votes
4 answers
370 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, ...
Emilio Pisanty's user avatar
1 vote
1 answer
1k views

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 ...
1 vote
2 answers
732 views

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 ...
Silver's user avatar
  • 706
2 votes
0 answers
277 views

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\...
faero's user avatar
  • 483
0 votes
1 answer
2k views

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)=\...
Amrat Butt's user avatar
3 votes
1 answer
1k views

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 ...
bolbteppa's user avatar
  • 4,101