Beta functions- Beta Gamma Formulas

Beta functions

A beta function is a special kind of function which we classify as the first kind of Euler’s integrals. The function has real number domains. We express this function as B(x,y) where x and y are real and greater than 0.

The beta function is also symmetric, which means B(x, y) = B(y ,x). The notation used for the beta function is “β”. The beta function in calculus forms an association between the input and output sets in integral equations and many more mathematical operations.

The beta function formula is as follows:

$B\left(p,q\right)=\underset{0}{\overset{1}{\int }}{t}^{p–1}\left(1–t{\right)}^{q–1}dt$

Beta Function Properties:

The following are some useful beta function properties that one should keep in mind:

• The beta function is symmetric which means the order of its parameters does not change the outcome of the operation. In other words, B(p,q)=B(q,p).
• $$B(p, q+1) = B(p, q)\cdot \frac {q}{(p+q)}$$
• $$B(p+1, q) = B(p, q)\cdot \frac {p}{(p+q)}$$
• $$B (p, q)\cdot B (p+q, 1-q) = \frac {π}{p \sin (πq)}$$

Some properties of beta function by a simple change of variables as

Example:

$\text{Evaluate the following:}\underset{0}{\overset{1}{\int }}{x}^{10}\left(1–x{\right)}^{9}dt$

Solution:

The given beta function is

$\underset{0}{\overset{1}{\int }}{x}^{10}\left(1–x{\right)}^{9}dt$

The above expression can also be as follows:

$\underset{0}{\overset{1}{\int }}{x}^{11–1}\left(1–x{\right)}^{10–1}dt$

Now, comparing with the standard beta function.

$B\left(p,q\right)=\underset{0}{\overset{1}{\int }}{t}^{p–1}\left(1–t{\right)}^{q–1}dt$

So, we can say that p = 11 and q = 10.

Using the factorial formula of beta function we get

$B\left(p,q\right)=\frac{\left[\left(p-1\right)!\left(q-1\right)!\right]}{\left(p+q-1\right)!}$

Here,

$p!=p·\left(p–1\right)·\left(p–2\right)\cdots 3·2·1$
$B\left(p,q\right)=\frac{\left(10!·9!\right)}{20!}$
$B\left(p,q\right)=0.0000005413$

Therefore, the beta function is 0.0000005413.