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Shilin, I. A.; Choi, J.

Method of continual addition theorems and integral relations between the Coulomb functions and the Appell function \(F_1\). (English. Russian original) Zbl 1522.35487

Comput. Math. Math. Phys. 62, No. 9, 1486-1495 (2022); translation from Zh. Vychisl. Mat. Mat. Fiz. 62, No. 9, 1522-1531 (2022).
Summary: The paper considers a function \(A\) introduced by the authors, which depends on one complex variable, two real variables, and one more argument, which defines a trivial or proper subgroup of a three-dimensional proper Lorentz group, which, therefore, is a real number or a pair of real numbers. In this case, the first three arguments define representation spaces and basis functions in these spaces. It is shown that its particular values can be expressed via the Coulomb wave functions or Appell’s hypergeometric function \({{F}_1}\). The resulting formula for the transformation of the function \(A\) is used to derive a continual addition theorem for this function and calculate the one-dimensional Fourier-Mellin-type integral transforms of the product of two Coulomb functions; its result is expressed via the function \({{F}_1}\).

Cited in 4 Documents

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35Q40 PDEs in connection with quantum mechanics
33C65 Appell, Horn and Lauricella functions
78A35 Motion of charged particles
78A10 Physical optics
81U05 \(2\)-body potential quantum scattering theory
35A22 Transform methods (e.g., integral transforms) applied to PDEs

Keywords:

Coulomb wave functions; Appell function \({{F}_1}\); three-dimensional proper Lorentz group; group representation; kernel of integral operator; Fourier-Mellin-type integral transform
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References:

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