The beta function (also known as Euler's integral of the first kind) is important in calculus and analysis due to its close connection to the gamma function, which is itself a generalization of the factorial function. Many complex integrals can be reduced to expressions involving the beta function.
The beta function, denoted by , is defined as
This is also the Euler's integral of the first kind.
Symmetry of the Beta Function
Because of the convergent property of definite integrals
so we can rewrite the above integral as
Thus, we get that the beta function is symmetric,
For positive integers and , we can define the beta function as
Recall the definition of gamma function
Now one can write
Then we can rewrite it as a double integral:
Applying the substitution and we have
Using the definitions of gamma and beta functions, we have
If we go by the definition of beta function to compute , we will have to solve the following integral
which is a very tedious work. Here is when the relation of beta function with gamma function comes in handy:
Using the substitution and , we have
The given integral is simply
Using the substitution therefore,
And by using the alternate definition of the beta function we can rewrite the above integral to fit the definition
then by switching to gamma functions
The recurrence relation of the beta function is given by
From the above relation and because of symmetry of the beta function, the following two results follow immediately:
We observe the reciprocal of central binomial coefficient:
This is a really useful relation, especially when solving summations.
Using the relation to the beta function, this is
We can interchange the summation and integral signs due to absolute convergence of the integrand:
The geometric progression sum was used. Now we can easily solve the indefinite integral and then put in the limits. So we have
The derivative of the beta function is a great way to solve some integrals:
Let's see how to use this, but first read the Digamma Function wiki.
Differentiate with respect to first and then to get
Put and to have
Using Stirling's formula, we can easily define the asymptotic approximation of the beta function as
for large and large
In probability theory, the beta distribution, written as for , is defined by the density
where and = as defined by the beta function.
is also known as the normalizing constant because it makes the integral equal to 1.
The -order statistic of i.i.d. Uniform distributions is modeled by .
The beta function can be implemented in Mathematica as follows: