A probabilistic proof of some integral formulas involving incomplete gamma functions

Robert E. Gaunt

Communications in Statistics - Theory and Methods, 2025, vol. 54, issue 8, 2246-2250

Abstract: The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals ∫0∞x−2νcos⁡(bx)γ(ν,αx2)dx (for ν>1/2, b > 0, α > 0) and ∫0∞x2ν−1cos⁡(bx)Γ(−ν,αx2)dx (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.

Date: 2025
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DOI: 10.1080/03610926.2024.2363870

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