Beta functions are a special type of function, which is also known as Euler integral of the first kind. It is usually expressed as B(x, y) where x and y are real numbers greater than 0. It is also a symmetric function, such as B(x, y) = B(y, x). In Mathematics, there is a term known as special functions. Some functions exist as solutions of integrals or differential equations.

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. We know that there are two types of Euler integral functions. One is a beta function, and another one is a gamma function. The domain, range or codomain of functions depends on its type. In this page, we are going to discuss the definition, formulas, properties, and examples of beta functions.

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 integral. 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.

The beta function in Mathematics explains the association between the set of inputs and the outputs. Each input value the beta function is strongly associated with one output value. The beta function plays a major role in many mathematical operations.

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 beta function plays a major role in calculus as it has a close connection with the gamma function, which itself works as the generalisation of the factorial function. In calculus, many complex integral functions are reduced into the normal integrals involving the beta function.

Relation with Gamma Function

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

Also, see:

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 B_z(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 can also be 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 value of the given expression using beta function is 1/ 280

Beta Function Applications

In Physics and string approach, the beta function is used to compute and represent the scattering amplitude for Regge trajectories. Apart from these, you will find many applications in calculus using its related gamma function also.

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