Questions tagged [special-functions]

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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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1answer
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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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1answer
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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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1answer
96 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 ...
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1answer
113 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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1answer
815 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}(\...
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1answer
141 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[\...
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1answer
26 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 ...
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1answer
91 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.
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2answers
146 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 ...
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1answer
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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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1answer
239 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 ...
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0answers
720 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 ...
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1answer
329 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) $...
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1answer
151 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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1answer
744 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 ...
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1answer
449 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 ...
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230 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\...
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1answer
908 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)=\...
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1answer
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 ...
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1answer
422 views

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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2answers
1k views

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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3answers
2k views

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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1answer
1k views

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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1answer
231 views

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}...