Beta Function

In Mathematics, there is a term known as special functions. Some functions exist as solutions of integrals or differential equations. We know that there are two types of Euler integral functions. One is a beta function, and another one is a gamma function. In this page, we are going to discuss the definition, formulas, properties, and examples of beta functions.

What are the Functions?

Functions play a vital role in Mathematics. It is defined as a special association between the set of input and output values in which each input value correlates one single output value.

Example:

Consider a function f(x) = x2 where inputs (domain) and outputs (co-domain) are all real numbers. Also, all the pairs in the form (x, x2) lie on its graph.

Let’s say if 2 be input; then we would get an output as 4, and it is written as f(2) = 4. It is said to have an ordered pair (2, 4).

Beta Function Definition

The beta function is a unique function where it is classified as the first kind of Euler’s integrals. The beta function is defined in the domains of real numbers. The notation to represent the beta function is “β”. The beta function is meant by B(p, q)

Where the parameters p and q should be real numbers.

Beta Function Formula

The beta function formula is defined as follows:

\(B (p, q)=\int_{0}^{1}t^{p-1}(1-t)^{q-1}dt\)

Where p, q > 0

The given beta function can be written in the form of gamma function as follows:

\(B (p, q)=\frac{\Gamma p.\Gamma q}{\Gamma (p+q)}\)

Where the gamma function is defined as:

\(\Gamma (x)=\int_{0}^{\infty }t^{x-1}e^{-t}dt\)

Also, the beta function can be calculated using the factorial formula:

\(B (p, q)=\frac{(p-1)!(q-1)!}{(p+q-1)!}\)

Where, p! = p. (p-1). (p-2)… 3. 2. 1

Beta Function Properties

The important properties of beta function are as follows:

  • This function is symmetric which means that the value of beta function is irrespective to the order of its parameters, i.e B(p, q) = B(q, p)
  • B(p, q) = B(p, q+1) + B(p+1, q)
  • B(p, q+1) = B(p, q). [q/(p+q)]
  • B(p+1, q) = B(p, q). [p/(p+q)]
  • B (p, q). B (p+q, 1-q) = π/ p sin (πq)
  • The important integrals of beta functions are:
    • \(B (p, q)= \int_{0}^{\infty }\frac{t^{p-1}}{(1+t)^{p+q}}dt\)
    • \(B (p, q)= 2\int_{0}^{\pi /2 }sin^{2p-1}\theta cos^{2q-1}d\theta\)

Incomplete Beta Functions

The generalized form of beta function is called incomplete beta function. It is given by the relation:

\(B (z:a,b)= \int_{0}^{z} t^{a-1}(1-t)^{b-1}dt\)

It is also denoted by Bz(a, b). We may notice that when z = 1, the incomplete beta function becomes the beta function. i.e. B(1 : a, b) = B(a, b). The incomplete beta function has many implementations in physics, functional analysis, integral calculus etc.

Beta Function Examples

Question:

Evaluate: \(\int_{0}^{1}t^{4}(1-t)^{3}dt\)

Solution:

\(\int_{0}^{1}t^{4}(1-t)^{3}dt\)

The above form is also written as:

\(\int_{0}^{1}t^{5-1}(1-t)^{4-1}dt\)

Now, compare the above form with the standard beta function: \(B (p, q)=\int_{0}^{1}t^{p-1}(1-t)^{q-1}dt\)

So, we get p= 5 and q = 4

Using the factorial form of beta function: \(B (p, q)=\frac{(p-1)!(q-1)!}{(p+q-1)!}\), we get

B (p, q) = (4!. 3!) / 8!

= (4!. 6) /8! = 1/ 280

Therefore, the beta function is 1/ 280

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