# Questions tagged [special-functions]

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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 ...
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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### 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 ...
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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### 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 ...
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): <...
79 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 ...
145 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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### 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 ...