Evaluation of a Double Integral with Exponentials and Two Generalized Laguerre Polynomials and Its Associated…

Abstract

We analytically evaluate in terms of finite sums, the following definite improper integral, which appears in the quantum field theoretical calculation involving elementary charged particles having charge ± e in presence of a constant external magnetic field B=Bẑ I_abln^μ₁…μ_r(q_⊥,|eB|) = ∫ d²k_⊥/(2π)²k_⊥^μ₁ … k_⊥^μ_r exp (−(k_⊥²+p_⊥²)/|eB|) L^a_l (2k_⊥²/|eB|) L^b_n (2p_⊥²/|eB|)|_{p_⊥=(q_⊥+k_⊥)} where μ_i ∈ {1,2}, l≥ 0, n≥ 0 and L_l^a(x) are the generalized Laguerre polynomials. We also provide explicit expressions of the result up to the fourth moment i.e. r=4, which are sufficient for most of the field theoretical calculations.

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BibTeX RIS

BibTeX

@misc{ghosh2026evaluation,
  title = {{Evaluation of a Double Integral with Exponentials and Two Generalized Laguerre Polynomials and Its Associated Moments}},
  author = {Snigdha Ghosh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35995},
  eprint = {2610.35995},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35995},
}

RIS

TY  - UNPB
TI  - Evaluation of a Double Integral with Exponentials and Two Generalized Laguerre Polynomials and Its Associated Moments
AU  - Snigdha Ghosh
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35995
UR  - https://arxiv.org/abs/2610.35995
AB  - We analytically evaluate in terms of finite sums, the following definite improper integral, which appears in the quantum field theoretical calculation involving elementary charged particles having charge ± e in presence of a constant external magnetic field B=Bẑ I_abln^μ_1…μ_r(q_⊥,|eB|) = ∫ d^2k_⊥/(2π)^2k_⊥^μ_1 … k_⊥^μ_r exp (−(k_⊥^2+p_⊥^2)/|eB|) L^a_l (2k_⊥^2/|eB|) L^b_n (2p_⊥^2/|eB|)|_p_⊥=(q_⊥+k_⊥) where μ_i ∈ {1,2}, l≥ 0, n≥ 0 and L_l^a(x) are the generalized Laguerre polynomials. We also provide explicit expressions of the result up to the fourth moment i.e. r=4, which are sufficient for most of the field theoretical calculations.
ER  -

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