A probabilistic proof of some integral formulas involving incomplete gamma functions
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909211877703680 |
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| author | Gaunt, Robert E. |
| author_facet | Gaunt, Robert E. |
| contents | The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals $\int_0^\infty x^{-2ν}\cos(bx)γ(ν,αx^2)\,\mathrm{d}x$ (for $ν>1/2$, $b>0$ $α>0$) and $\int_0^\infty x^{2ν-1}\cos(bx)Γ(-ν,αx^2)\,\mathrm{d}x$ (for $ν>0$, $b>0$ $α>0$), where $γ(a,x)$ and $Γ(a,x)$ are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10004 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A probabilistic proof of some integral formulas involving incomplete gamma functions Gaunt, Robert E. Probability Primary 33B20, Secondary 60E05 The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals $\int_0^\infty x^{-2ν}\cos(bx)γ(ν,αx^2)\,\mathrm{d}x$ (for $ν>1/2$, $b>0$ $α>0$) and $\int_0^\infty x^{2ν-1}\cos(bx)Γ(-ν,αx^2)\,\mathrm{d}x$ (for $ν>0$, $b>0$ $α>0$), where $γ(a,x)$ and $Γ(a,x)$ are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas. |
| title | A probabilistic proof of some integral formulas involving incomplete gamma functions |
| topic | Probability Primary 33B20, Secondary 60E05 |
| url | https://arxiv.org/abs/2309.10004 |