# Beta function

Beta functions are a special sort of function, which is additionally referred to as Euler integral of the primary kind. It’s usually expressed as B(x, y) where x and y are real numbers greater than 0.

#### Input

$$a=1.5$$ $$b=0.4$$

#### Solution

$$B(a,b) =2.0439411002255$$

#### For Example :

$$\normalsize Beta\ function\ B(a,b)$$ $$(1)\ B(a,b)= {\large\int_{\tiny 0}^ {\tiny 1}}t^{a-1}(1-t)^{b-1}dt,$$ $$\hspace{165px} Re(a)\gt 0,\ Re(b)\gt 0$$ $$(2)\ B(a,b)={\large\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}}\\$$

## What is Beta function and what are the uses of it?

Beta functions are a special sort of function, which is additionally referred to as Euler integral of the primary kind. It’s usually expressed as B(x, y) where x and y are real numbers greater than 0. it's also symmetric function, like B(x, y) = B(y, x). In Mathematics, there's a term referred to as special functions. Some functions exist as solutions of integrals or differential equations.

## What are the Functions?

Functions play an important role in Mathematics. It’s defined as a special association between the set of input and output values during which each input value correlates with one single output value. We all know that there are two sorts of Euler integral functions. One may be a beta function, and another one maybe a gamma function. The domain, range, or codomain of functions depends on its type. During this page, we are getting to discuss the definition, formulas, properties, and samples 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 within the form (x, x2) lie on its graph.

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

## Beta Function Definition

The beta function may be a unique function where it's classified because the first quite Euler’s integrals. The beta function is defined within the domains of real numbers. The notation to represent the beta function is “β”. The beta function is supposed 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 therefore the outputs. Each input value of the beta function is strongly related to one output value. The beta function plays a serious role in many mathematical operations.

Beta Function Formula

The beta function formula is defined as follows:

B (p,q)=∫10tp−1(1−t)q−1dt

Where p, q > 0

The beta function plays a serious role within the calculus because it features a close reference to the gamma function, which itself works because of the generalization of the factorial function. In calculus, many complex integral functions are reduced into the traditional integrals involving the beta function.

## Relation with Gamma Function

The given beta function is often written within the sort of gamma function as follows:

B (p,q)=Γp.ΓqΓ(p+q)

Where the gamma function is defined as:

Γ(x)=∫∞0tx−1e−tdt

Also, the beta function is often calculated using the factorial formula:

B (p,q)=(p−1)!(q−1)!(p+q−1)!

Where, p! = p. (p-1). (p-2)… 3. 2. 1

### Beta Function Examples

Question: Evaluate: ∫10t4(1−t)3dt

Solution:

∫10t4(1−t)3dt

The above form is additionally written as:

∫10t5−1(1−t)4−1dtNow, compare the above form with the quality beta function: B(p,q)=∫10tp−1(1−t)q−1dt

So, we get p= 5 and q = 4

Using the factorial sort of beta function: B(p,q)=(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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### Q. What Is Function?

A. A Function Is A Binary Relation Between Two Sets That Associates Every Element Of The First Set To Exactly One Element Of The Second Set.