This article explores Euler integrals, specifically the gamma and beta functions, and their significance in mathematical analysis, with applications in differential equations, probability theory, and statistics. The gamma function, defined as a continuous extension of the factorial, is analyzed for its key properties, including continuity, infinite differentiability, and the functional relation Γ(z+1)=zΓ(z). Singular points (t=0 and t=∞) and the uniform convergence of the integral are examined using the Weierstrass criterion. Applications of the gamma function in probability distributions, quantum mechanics, and statistical physics are highlighted through examples. The article also includes a practical integral calculation and is supported by references to mathematical literature.
Tog'ayev Turdimurod Xurram ug'li, Safarova Durdona Muzaffar qizi, Eshtemirov Eshtemir Salim oʻgʻli, (Mon,) studied this question.