Beta Function: Overview, Questions, Preparation

Beta functions are a special kind of function often regarded as the first Euler integral. It is commonly represented as B(x, y), where the real numbers x and y are greater than 0. It is also a symmetrical function. For example, B(x, y) = B (y, x). There is a concept defined as unique functions in mathematics. As solutions of integrals or differential equations, certain functions occur.

What are functions?

In mathematics, functions are critical. They are characterized as a particular relationship between input and output (a collection of values) in which a single output value is associated with each input value. We know there are two kinds of integral Euler functions. A beta function is one, while a gamma function is another. The function domain, range, or codomain depends on its form. 

For instance, consider a function f(x) = x2 where all real numbers are inputs (domain) and outputs (co-domain). 
If 2 is the input, we will get 4 as the output, and it's written as f(2)=4. It is said this is an ordered pair (2, 4).

The Formula of Beta Function

The formula for beta functions is:
B(p,q)=∫10tp−1(1−t)q−1dt
p, q > 0
In calculus, the beta function plays a significant role since it has a strong relationship with the gamma function, which itself behaves as the generalization of the factorial function. In calculus, the usual integrals that include the beta function are reduced to many complex integral functions.

Definition of the Beta Function

The beta function is a special function under which the first form of integrals of Euler is classified. The beta function is specified in real-number domains. "The notation for the beta function to be defined is "β". The beta function is referred to as B(p, q), where the p and q parameters have to be real numbers.
In mathematics, the beta function describes the relationship between the input set and the outputs. The beta function closely correlates one output value with each input value. In several mathematical activities, the beta function plays a significant role.

Applications of the Beta Function

The beta function is used in the physics and string method to calculate and represent the scattering amplitude for Regge trajectories. Besides these, in calculus, you can often find several implementations using the related gamma function.

This concept is taught under the chapter Relations and Functions in class 12. In this chapter, you will learn about the various types of functions and apply this knowledge to solve questions. The weightage of this chapter is eight marks. 

1. Solve ∫10t4(1−t)3dt
Solution.

∫10t4(1−t)3dt
∫10t5−1(1−t)4−1dt
B(p,q)=∫10tp−1(1−t)q−1dt
We get p= 5 and q = 4
Using the factorial form of beta function: B(p,q)=(p−1)!(q−1)!(p+q−1)!
B (p, q) = (4!. 3!) / 8!
= (4!. 6) /8! = 1/ 280
Therefore, the beta function is 1/ 280

Q: What is the use of the beta function?

Q: How do you solve a beta function?

Q: What is the beta distribution used to represent?

Q: What is the beta probability distribution?

Q: How do you find the beta of a function?