The beta function is a special mathematical function denoted as B(x, y) . It is defined for positive real numbers x and y and is expressed in terms of the gamma function: B(x, y) = \frac\Gamma(x) \Gamma(y)\Gamma(x+y) . The beta function is commonly used in calculus, particularly in the evaluation of integrals and in probability theory. In addition to its mathematical significance, the beta function has applications in various fields, including statistics and physics. It often appears in the context of probability distributions, such as the beta distribution, which is useful for modeling random variables limited to a finite interval.