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.

## Practical Uses

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.

## History

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.

## References

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

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