The Beta function (also called the Euler Integral of the first kind) is a definite integral, related to the gamma function. The most common representation for the function is:
The function goes by many different names. As is is usually defined by the above integral, it is sometimes called the “beta integral.” It’s also called the Euler Β-function, and is sometimes simply denoted by its variables: Β(p, q).
Other equivalent forms of the function exist, and are obtained by changing variables. You can find a list of various forms (including trigonometric forms) in the Digital Library of Mathematical Functions.
The function is used in physics and string theory, where it can model properties of strong nuclear force. In time management problems, the beta distribution (which is the integrand of the Beta function) can be used to estimate average time to complete tasks. In calculus, the function is especially important because many other definite integrals can be rewritten as a Beta function: this useful general function can simplify the evaluation of many complicated integrals.
In probability and statistics, the Beta function has many applications. It is sometimes used as a normalizing constant or as part of some definitions for probability distributions. For example, the probability mass function (PMF) for the Yule-Simon distribution uses the Β-function for its PMF. The Beta function can also define a binomial coefficient, after adjusting indices. These statistical applications extend to a wide variety of real life applications, including modeling experimental frequency distributions of relative sunshine and humidity.
While Euler first developed the beta function, it was the French mathematician Jacques P.M. Binet who first used the beta symbol for the function.
Other Related Functions
The incomplete beta function is a generalized Β-function, where the definite integral is replaced with an independent integral.
Alhassan, E. et al. (2015). On some Applications of Beta Function in some Statistical Distributions. Researcher.
Riddhi, D. Beta Function and its Applications. Retrieved December 5, 2019 from: http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf
Factorial, Gamma and Beta Functions. Retrieved December 6, 2019 from: http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap1.pdf
Stephanie Glen. "Beta Function: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/beta-function/
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